You are on the right track. First see if it moves better after you loosen the screw. You might start with an 1/8th or 1/4th turn. If it turns well and doesn't slip after reaching the proper pitch. If that doesn't work, tighten the screw until the string holds under tension. Now try lubricating the tuner with either graphite or a dry silicone spray. I prefer the first as the silicone spray can get on the finish and harm it. You may need to turn the peg completely around many times. Oh, you could also clean the peg with either a commercial degreaser and/ or denatured alcohol. This best done with tuners off of the dulcimer. Hope this helps.

Ken

"The dulcimer sings a sweet song."

The problem is likely related to the vintage of the dulcimer. That is a long time and things can dry out over time.

I have taken some similar tuners apart. On the one I had apart grease had dried over time behind the gear. And the gear was dragging a good bit on the screw going through it. I took it apart, cleaned everything up and reassembled with a very light coat of light sewing machine oil. The tightness of the screw through the gear plays a part in the tuner holding its place.

I helped a friend diagnosis his cheap foreign made dulcimer and in his case the worm threads on the shaft with knob were badly worn. That tuning machine needed to be replaced.

Replacements are available. I am not that familiar with  different brands. Some require different size holes through the wall of the peghead. Perhaps a local guitar center could help you  with parts if replacement is needed.

I play with a dulcimer club located in Fairhope, Al and today I was assisting a young lady in tuning her dulcimer.  She has a beautiful Folkcraft dulcimer which as an excellent voice.  The dulcimer is of the vintage when Folkcraft's company was located in Connecticut.  The issue is the instrument is hard to tune because the tuners are extremely hard to turn.  Attached please find a photo of the tuners for your reference.

I have no experience with these tuners so I thought I'd ask the experts before I screw them up.  My thoughts are I could loosen the screw going through the gear or possibly put graphite under the gear but my concern is doing something that would make the instrument unable to hold tune.  Your advice is appreciated.

Thanks in advance

Jim