You can reuse the Monster Clay. The mineral spirits will break down the surface making it almost muddy. Use it to smooth surfaces or to soften texture. The mineral spirits will eventually evaporate but I think with white it may take a few days.
I use a different oil based clay but I've found it can take a good few days to evaporate leaving a sticky & very delicate surface. I now use lighter fluid, it evaporates very quickly & with a dusting of talc after can give a relatively hard smooth surface.
Might be an idea to try different thinner's on a spare bit of clay to see what works best for you.
The amount you use matters a good deal. I use white spirits over monster clay and have never had a problem with it taking too long to dry out. I brush on a very light coat however which makes it workable for about ten minutes and dry in about thirty. The brand could be the reason why some spirits aren't drying quickly. I use the stuff from hobby lobby.