I can just see me putting on a frozen T-shirt. EEK! I could enter the "Wet T-Shirt" contest..and lose.
Even if over my other summer shirt it would get soaked. I guess it's a guy thing.
I tend to stay in or go from A/C in car to A/C in a store. I don't sweat, I can't cool down ,so I try not to get overheated.
If I do, I take a cold shower a few times or rub cold facecloth over my exposed parts. Good ideas btw.