Here in the desert, hats are important... they're sort of portable shade. And I don't care how dry the heat is, you're going to sweat at 100F+, so the issues you mention about damp headgear are familiar to me. The best solution I've found is a straw hat. They're light, and sweat-absorbant, and you can get them in a number of styles. Mine is in the shape of a cowboy hat, for instance. They're relatively inexpensive, so if you lose one, or the cat pees on it, or it just gets too grody, you don't have to feel guilty about disposing of it.