Ten year old strings? Those strings will be 'dead.' First change them, then see if the pegs are doing their job. I suggest you tune it DAA (not DAD) to reduce the pull from the melody string.

There are good-sounding suggestions here from kind mudcatters, but put new strings on first to see how they hold.

Where has the dulcimer been stored? Perhaps it simply needs a few weeks in a normal environment. As I'm sure you know, instruments like the same environment we do: temperature about 70 and humidity 40 - 60%. If the dulcimer was stored under a bed and near a heat register, it will have dried out, shrunk, and will not hold the pegs tightly.

I have a dulcie made from a kit, and it plays fine after more than ten years. I change strings about every nine months, but the highest string more often.