My guess would be that your burner is not fully combusting the gas, thus causing the soot. Can you adust the burner to make sure your flame is running blue with no or very little orange? SOunds like it needs more O2.
Also, we used to have this problem with white gas camp stoves that weren't burning correctly. I used to coat the bottom of the pot in liquid Dawn dish soap prior to cooking. The soot would come right off with a little water. However, I'm not sure if this would work for propane. I suspect it would but don't won't cause you any problems.