Wow, she looks really good! As a simple, temporary fix, you could loosen the strings and place a little slice of good grade of paper in the nut and see whether that fixes your buzzing. If it doesn't, you may just need another string gauge. (I'm not a luthier so my advice is just that of a strummer.)
Before sanding, you might want to try some Hill Peg Compound, which is a substance used by fiddle players to make the pegs turn more freely. I have used it successfully on a number of occasions to get a smoother and more accurate tuning with wooden tuning pegs. It comes in a tube (like chapstick) and is not expensive considering one tube will probably last you a lifetime. Just rub a little on the tuning peg where it makes contact with the peghead. You can purchase some on E-Bay if your local music store doesn't carry it.
If you're going to sand the pegs, use a VERY fine sandpaper -- 800 or 1000 grit. A TINY change in diameter makes a huge difference.
Better yet, take the pegs and dulcimer to a violin shop and ask them to correct the peg fit.
updated by @ken-hulme: 09/21/19 08:52:16AM
I was also thinking of something like Grover Champion friction tuners if I did buy some, maybe they would look better.
I have no idea who made this dulcimer or where it was made, I wish I did. I bought it on ebay out of Alford, Great Britain. There are no markings on it as to who made it, and the top and bottom are made of plywood.
Another option is to get some Banjo tuners. I did put on some ukulele tuners on a small dulcimer and it looks good on that one....but your dulcimer here looks pretty big and to balance out the size of things, you might consider banjo tuners that are pictured in the above photo as you open this site. I just finished dulcimer #29 today and these are what I used. I like them very much. If you get them straight from China...cheap...and you don't know what you're getting. If you get them from a music store, if they don't work like you want them to...can take 'em back. I ordered them from a place in Kentucky...got 'em in 4 days. happy about that. $59.00. yep, you'd have to enlarge the holes. Where is this dulcimer made? aloha, irene
No reason they won't work. You may have to enlarge the holes a bit -- it depends on the diameter of the shaft relative to the diameter of the hole. Check the specs on those Grover 8Ns for shaft size.
A few minutes work would have made those wooden tuners work just right...
Will Grover 8N geared Ukulele tuners work alright on a Dulcimer, before I go ahead and order any? I want to put them on that dulcimer I just bought, it has these infernal violin tuning pegs that are all seized, probably because they don't fit properly, they aren't ebony, and it's been really humid. Got 2 out of 3 out, but it took several days and a lot of patience with what I have to work with...