How to keep thin sponge bases moist?
Is there a reason you're not just baking a standard thickness and splitting it into layers?  This is going to give you more control.

If you have troubles moving the layers without breaking them, have a piece of cardboard handy (traditionally a cardboard cake round would be used by any would work) and after cutting the cake horizontally, lift up the edge of the top piece to slide the cardboard under and then proceed to push it all the way under the top layer before carefully removing the layer.  Proceed with the next layer.

If they cake is a bit dry but not crunchy, moisten the cut surface by brushing with simple syrup on its own if you don't want any added flavor, or add a bit of a complementary-flavored liqueur to the syrup before moistening.