A few paper towels soaked in water to sponge off with plus a small hand towel from home for drying work fine for me. I shower before I leave, so there isn't much of a problem.
Arriving early enough to slowly walk around to cool off before getting dressed helps.
I find that sweat is mainly a problem in the winter, oddly enough. If I'm wearing enough clothing to keep warm at -22 C, it's enough to make me start sweating.
Underwear choices are very much a YMMV situation. I love the modern wicking fabrics and never have a problem with odor. Perhaps different people have different sweat chemistry. Damp wool smells like a wet sheep on me.