"Could they have purposely installed an actual nut to bypass the zero fret?"
Yes. The problem with that is the locations of each of the other frets is based on calculations using the VSL [Virtual String Length] from the center of the zero fret to the nut side of the saddle.
We had a family emergency so I will not be able to do anything with this thing until next week, but all of this is good information! I also remembered that I have a Dremel tool that may come in handy although the manual hand files may be a little gentler on the wood and plastic. I will agree with some of you that somebody has worked or modded this dulcimer. Could they have purposely installed an actual nut to bypass the zero fret? I also did not notice that the saddle also looks like it has been altered. Right now, I have the one that came with it removed and have a bone one made for a guitar in its place. I have sanded it down to lower the string action and I have a few of them in case I mess it up! I was going to sand the original one but of course, do not have a spare. One way or another I will get this instrument fixed up. Thanks again for all the help!
I went back to look at the photos of the spacer and fret. Someone obviously messed it up.
Frankly, if I was dealing with this, I would remove the plastic spacer and throw it away. I would make a new spacer from hardwood or plastic. I would probably then cut a new groove to fit the width and height of the spacer closer to the zero fret. Alternately I would re-cut the spacer hole to square it up, glue in a filler block, and then cut a new slot there to fit my new spacer.
I would also reset the position of the middle string to the middle of the fretboard.
Sorry about misleading you about sloping the grooves. That applies to a nut, not a spacer.
You want to angle the slot so that the high point is on the bridge side to preserve the VSL.
I have a couple of needle file sets, neither was expensive nor diamond coated. They do the job. You can find them at the two places Wally suggested plus Lowes and Home Depot. I've used the welding tip cleaners mainly to expand slots because of binding strings, but not to make new slots. I have a set of Stewmac nut files that I use to start the slots.
Ken
"The dulcimer sings a sweet song."
I will get a set of those needle files. We are going to Walmart today and I will see if they have them. I bet I have some somewhere but have no idea where they are! I have a few sets of those welders tip cleaners and tried them last night on the A string slot and they just did not work.
You don't need or want diamond files for cutting wood or plastic. I use cheap (under $5) "needle" or "jewelers" files like these.
https://www.harborfreight.com/needle-file-set-12-piece-56743.html
Unless, of course, you are using something like stainless steel for your nut.
Some recommend using welder's "Tip Cleaners." They provide a round bottom groove. I've tried them, but I think a ve-bottom is better.
I am actually looking at some diamond file sets made specifically for doing this, but most seem to be real expensive. I see a set on Amazon that I would love to get but would run me 120 dollars. If I was a professional, I would not hesitate to get these, but they will not be used often. I do not want the get the cheapest thing out there either. Do any of you know a good set of diamond files that will be good for filing the slots that is a bit cheaper??
To make the slot adjustment, I would use either a triangular needle file or perhaps a welding tip cleaning file. You can do it one string at a time by slacking the string and pulling it to one side while you file while you keep the others tight.
You want to angle the slot so that the high point is on the bridge side to preserve the VSL.
I would NOT "Cut" them with a saw.
I think I will try cutting the slots deeper in the "nut". I really do not want to ruin the dulcimer as I do not have very good woodworking skills! I probably should have invested in a new better brand instrument, but I am just a beginner at this and did not want to spend a whole lot of money. Knowing me though if I saw a good deal on a better one, I would probably buy it! The one I got from Amazon is really nice though, but I know it is nothing high end. I did a little research on the shop that is named on the sticker inside the sound hole. Apparently, the phone number is still good, and the shop may still be in business, but the instruments designer, Jerry Meador passed away in 2018 at the age of 68. I am going to guess this dulcimer was probably made in the 1980's. I was able to find a few other photos online of the same model, and one had a black "nut" and the other was like mine, but I could not tell where the strings were in relation to it. I will keep you all posted of any progress I make!
I agree with Skip about cutting the slots deeper in the false nut before trying anything else. Sometimes I get ahead of myself and forget to do the easier solutions first.
Ken
"The dulcimer sings a sweet song."
I think I would try cutting the slots in the false nut deeper as a work around for now. Be sure to protect the wood.
Once you get the nut problem fixed the error should be minimal.
An example of splitting the error, Say the note/fret you check is sharp by 10 cents, you can re tune the string to 5 cents flat then when the string is pressed it will only be 5 cents sharp at that fret. The caveat is this affect all the notes/frets. If you have to fudge a bit, a little sharp is more acceptable than a little flat.
Just a thought, but have you tried contacting the makers/builders? your photo of the label shows a phone number and address. they could give you info on the instrument, like if that blue plastic nut-like thing was part of the original dulcimer or added by a former owner, maybe some hints on how to remove it safely, etc. It can't hurt to ask...
Mike, that blue piece is not actually a nut (or shouldn't be). It should be a string spacer to keep the strings in the proper place across the nut which is the zero fret. If it were my instrument I would try to get the blue plastic piece off and replace it with a small piece of wood; maybe one eighth inch square, or perhaps three sixteenths or a quarter inch square depending upon what would look good. I'd saw slots for the strings to provide for double melody strings and four equidistant string spacing. I'd make the slots deep enough so that the strings rest on the first fret. After that, I'd tackle the string height from the saddle/bridge end.
Ken
"The dulcimer sings a sweet song."
OK, all of this is great information, and I have already learned quite a bit! I did not know what the zero fret was before you all just told me and was wondering why that fret was doing there! I have not tried the trick with the nickel yet but just took a look at the zero fret and strings and posted more pics. The strings do not come close to touching that zero fret even when fingering the strings on fret 1. I am just guessing, but looking at the nut on this instrument it looks like the previous owner may have added it to bypass the zero fret for one reason or another. It was unfortunately glued on, and I do not have the skills to remove it without damaging my dulcimer. Later on today I am going to do as you all suggested and check the string height on the 7th fret and see if I can lower the action by sanding down the saddle some more. I do have some files to deepen the nut slots but have never used them.
One other tip is to split the error between # and b
Sorry Skip but I do not know what you mean by this! 
More to come and again and I thank all of you for the information you have given me so far!
Skip and Nate have given you good advice. Check to make sure that the zero fret is higher than the first fret after you check to see that all strings are resting on the zero fret. From there check with the nickel on the seventh fret. If you get lower than that, the strings may start to buzz on the frets. You can make slight adjustments at the saddle (bridge) if you need to lengthen the vibrating string. You can't shorten the string because the saddle is in a fixed position.
Ken
"The dulcimer sings a sweet song."
First, welcome to FOTMD.
Are the strings touching the fret right next to the nut? If not, you need to cut the slots deeper because that fret is the zero fret. A zero fret performs the same function as a nut on a guitar. That black piece looks like nut but is just a string spacer.
There are three main factors to consider when intonating a dulcimer, the height of the nut, the height of the bridge, and the position of the bridge.
There is a simple system for intonating a dulcimer which is called the "nickel and dime method" where a dime is placed on the fingerboard just above the first fret so that the side of the dime is touching the nut-facing side of the first fret, so the dime is between the first fret and the nut. A nickel is placed on top of the 7th fret and is between the 7th fret and the string.
Generally, the nut and bridge should be lowered until the string is just barely touching the face of both of the dime and the nickel.
If your fret wire is taller than average, the dime may not be useful, since the fretwire may be taller than the dime. If that's the case, it may be a little bit more complicated. Using a dime only works for short fret wire, such as a mandolin or ukulele fret wire, and if the instrument has guitar or jumbo fret wire, then determining correct intonation at the nut might be more difficult.
If you set the string height or "action" using this method and all goes well, but your instrument is still not well intonated, then you may need to adjust the position of the bridge itself.
To do this, tune the string up to the desired note, and then using a tuner check the intonation at the octave fret.
If the octave fret is sharper than the root note, then the bridge needs to be moved closer toward the nut. If the octave fret is flatter than the root note, then the bridge needs to be moved further away from the nut.
I hope this helps
Nate
First time poster and first time dulcimer owner! I recently purchased two dulcimers. One was an Applecreek AD200 cherry model from Amazon and I have been very happy with it. Its finish and sound quality exceeded my expectations. I also wanted a US made vintage model so I started looking on Ebay and there was a lot to choose from! The first one I purchased I returned because the seller put down the wrong dimensions for it, and it was too small for my taste, but sounded OK, I finally found one that was the size I was looking for, and it arrived a few days ago and that is where my question starts! It does not have a brand name on it, but a sticker inside the sound hole that states " The Beaten path, Home Of American Dulcimers" in Pigeon Forge, TN. It also shows two names, Jerry and Marsha Meador. I see no date or model number so I am assuming they made this instrument. If any of you know anything else about them or this dulcimer I would appreciate it also! Anyhow, the dulcimer was in pristine shape except for a crack in the headstock, but it does not appear to run very deep. I cleaned it up and put new strings on it and tuned it. It sounds great when just strumming an open cord, but hitting notes up on the frets sound off, especially with the center A string. I know how to intonate most electric guitars but am not sure how to do it on this. The saddle is not movable, so I am stuck with it as far as that goes. I removed it and replaced with a bone one made for an acoustic guitar, sanded it down a bit to lower the string action, and this has helped a little bit, but it is still off. What else can I do? Any and all help is welcome! I have attached some photos for reference.
