Hi David. I am no pro on this matter but here are a few tips, I've learnt:
1. Use a rattle-can of solvent based primer to seal the object. Test a scrap to see if the solvent attacks the foam. Solvent based clear polyurethane works, too.
1a. Acrylic paint is ideal on foam core because you do not need to use solvents like with oil paints. Acrylic it fast-drying, durable and adheres to most surfaces. It also has an even sheen and does not need a base coat.
2. If you don't need the foam board to be stiff, you can remove the paper by first soaking it with rubbing alcohol. This will diminish warping and allow you to bend it, shape it or apply it to a flat board. I usually just peel the backing off with a sharp hobby knife.
3. If you apply too much paint at once, the outer layer of paper on the foam gets saturated, and as it dries, the paper shrinks but the foam inside doesn't, and that cause the edges of the paper to curl and warp.
By you using balsa wood and painting it with water you run the risk of the wood soaking up the moisture thus warping the balsa easily, in turn warping the foam board. I think that's what happened in your case. I could be wrong.
I am sure there is more to add...Hope this helps.