Welcome Syd;
The shape of the dulcimer isn't the problem. No shape, except a box, is any more stable than another. Chances are it's
A) the 'slippery' pants/shorts you wear. Use that rubbery shelf liner material as a mat across your lap.
or
2) your basic 'posture' with the instrument. Many beginners try to sit too prim and proper. Start with your knees wide apart - more than 18 inches! Sit upright, not laid back, not hunched over. Then don't put the dulcimer at right angles across your lap. Angle it so that the right end is tucked back by your right hip and the left end is out over your left knee - about the 1-3 frets over your knee. As you strum the right end will try to work its way out - on an instroke with your picking hand brucsh your little finger into the body, pushing it back against your hip.
If you practice good posture, you absol;utely do not need a strap or anything like that. If correct posture doesn't work, conside gettin/making a table of some sort on which you can put the dulcimer while you play.

But then neither am I