Advice on beginner’s dulcimer
Instruments- discuss specific features, luthiers, instrument problems & questions
Lisa, can you tell us the length, in inches, between the nut and the bridge? If it's 28" or longer, it might be a bit of a strain for those cheaper older tuners to hold the strings up to DAA or DAd. I have an old 1970s kit dulcimer that had tuners like yours, and the screw threads just didn't hold very well anymore. Perhaps if you get slightly lighter gauge strings, and/or tune down a step to CGG or CGC (instead of DAA or DAd) ...the tuners might be better at holding.
I wound up replacing the tuners on mine with some new (but using same sized holes) inexpensive tuners... that helped, but actually it didn't totally cure the issue for my kit dulcimer, alas. Mine remains a little frustrating to tune. Your mileage may vary.

lolol Never hurts to have a spare!