I try to use acrylic ink and acrylic paint in my Alumilite Clear Slow and it foams up and volcanoes on me. Is it because it's water based? Is there a way to keep that from happening, or is there a different kind of paint that is good for this particular resin? Should I just stick to Mica and glitter?
Yes, it is moisture that causes it to foam, and it is unavoidable. I don't know of any paint that will work. You can use alcohol based dyes and inks, or pretty much any dry powder (but don't expect a powder to dissolve).
Yes it is not recommended to use a water based paint with the Clear Slow as it adds moisture and moisture is not your friend with the Clear Slow.
That being said, acrylic paint is usually what is used with Amazing Clear - I've used a cheap acrylic craft paint in the Amazing Clear with no problem but Amazing Clear is actually an epoxy and Clear Slow is a resin. So they are two different types of resin which means they will react differently to different kinds of paint. What works for one may not work as well for the other.
LawnBoy has the right idea, you can try the alcohol based dyes and inks or powders which should work as they are non-water based and non-water based coloring types are what's recommended.