Help! New (old) Dulcimer w/ loose tuner knob
Instruments- discuss specific features, luthiers, instrument problems & questions
It seems you have a very common type of inexpensive tuner on your dulcimer. These are called friction or "Grover" pegs, and are used quite a bit on dulcimers. They work on the theory of squeezing against their mounting holes. The smaller the string gauge, the less they have to grip the hole. As you get to the wound bass string, more tension is required to stay put, thus more gripping action is needed. Usually all one needs to do is to tighten down the end screw until the peg grips well enough to stay where you put it.
I would take the damaged screw to the local hardware or home center near you, and show them what you have. Maybe somebody there could find another similar one for you. Or a guitar repair shop (such as Guitar Center) might be able to find a similar one that would work for you. The main thing is to be sure to reassemble the peg properly, with the little fiber washer in there that gets squeezed when you tighten the screw down. I hope these ideas will work for you. Good luck!