Almost forgot to answer your original question...
You've gotten some great answers so far, but the best advice I've ever read (can't remember source) is this: Rubbing Alcohol. Seriously.
First swab off the sweaty areas with a wet wash cloth or baby wipes, then apply Rubbing Alcohol.
I keep it in a tiny spray bottle & spritz on liberally! Another option is hand sanitizer with a high concentration of alcohol in it (this option seems stickier to me, so not my favorite.)
This does several things (1) helps cool down your skin to stop sweating quicker & (2) kills the bacteria that if left on can get funky.
After the rubbing alcohol dries, apply deoderant. I also sprinkle on some baby powder because I like the fresh scent & soft feel (I'm a such a girl.) Then dress in completely clean, dry clothes (I keep extra socks, bras, panties & tank tops in my locker to put on under hospital scrubs.)