What brand do you use at Disney? I thought I saw some sold there when we were there in February but I could be wrong (why would you need them in February with temps in the 70-80s?).
The waterparks sell them for sure but I think the parks sell fans and things more so than the cooling towels although you could probably still find them.
Also, yes you do. Even if it doesn't initially feel that hot running around the park and being around people all day under the bright sun will burn anyone out. Even cast members.