HeyJoe. I've been looking for other folks who have experienced similar symptoms to yours, but with no luck! The poison center is correct - it is not possible to be poisoned by mushrooms through skin.
Did you ever cut any of them in half? It would really help to know what species you even have, it's really next to impossible with just the pic you posted. Also, if you're deterred from shrooming because you broke the cardinal rule of mycophagy, never eat a shroom without 100% positively id'ing it as an edible species, then that decision is more than likely for the best! There are mushrooms that can kill you!! Hot hands are nothing compared to liver and kidney failure. If you had posted that pic of the unknown shrooms BEFORE you prepared them for the kitchen, you would have been told quickly that you were wasting your time!
BTW, my theory is that you touched a plant that causes contact dermatitis or you are allergic to. Did you pull any weeds that day or brush aside any unfamiliar plants while in the woods? I have had the symptoms you describe from such a situation, especially the flaming hands!