I've never reacted this quickly to the heat before, either. I'm sweating as soon as I move, even in AC...and I start to feel "weird" after being overheated for just a little while, then the locking type of spasms begin.
I tried using the neck cooler I got from the Copaxone people, but it doesn't work well. I'm considering looking around for a cooling vest. Otherwise, I'll be vegging the rest of the summer, and with DD's wedding coming at the end of July, that's not possible...