Adding more water to an already hot and humid situation risks a Wet Bulb.
If the water coming out of your cold tap is cool (which it should be, since pipes are typically underground), then I think maybe it would still work because you could refresh the t-shirt with colder water occasionally. It's just less than ideal compared to doing it in a dry climate.
That's not why you catch a cold. A cold virus is spread by rain, that's why you should avoid getting wet under the rain. Or go outside in the first place.