NSThoreau, I have the same exact same old Dulcimer Shoppe kit. It has a 28" scale length, and the tuners had a hard time holding the strings. I changed the tuners years ago, but my dulcimer still has a bit of a problem holding its strings at high tension. Not exactly sure why that was, but I wasn't about to spend another amount of $ on a third set of tuners, so I left it with my replacements, which are working ok. The instrument is made from quite thin wood.
If you continue with the problem, I might suggest you change to .010 strings for the melody strings rather than .012, since the thinner strings will be at a lesser tension. You also might want to try tuning to the key of C rather than D, which helps as well to reduce the pull on the strings.
Those irritated by grain of sand best avoid beach.
-Strumelia proverb c.1990