Well I THINK I solved my problem (bear in mind I am certainly not a chemist).
I read online that sulfur is practically insoluble in water, and although I'm not sure if clay is technically soluble in water, I thought I might try a make shift cold water extraction attempt. It worked!
I put 3-4 spoon fulls of the gardening sulfur (90% sulfur, 10% clay) in a plastic jar and than filled the rest with about 1.5 cups of warm water. I closed the lid and then shook it until the solution appeared to have the gardening sulfur completely suspended. I then put the jar in the refrigerator for about 2 hours and sure enough, the sulfur had fallen to the bottom of the jar to form a nice think layer while the remaining clay was still suspended (or dissolved, I'm not quite sure). I then opened the jar and slowly poured the liquid out, leaving the sulfur paste at the bottom (it pretty much just stuck to the bottom even when pouring). I then spooned it out, placed the paste on some newspaper, and let it dry out in the sun for a few hours. I used it in some simple rocket fuel and I can see a slight increase in thrust.
Anyway I thought I'd post that this might be a good way to concentrate the sulfur from gardening sulfur. Again, I am no chemist and this might be a dumb way to do this.