Hi A.,
I feel for you!! I used to sweat like a hog - and I do not use the term lightly. I found a product a few years back called Certain Dri. It is a miracle, I never, ever have dampness under my arms - nor any funky pit odor.
Different things people suggested to me in the past: put your deodorant on at night before bed, clean your pits a few times a day with rubbing alcohol and or tea tree oil, wear cotton.
I have shared with a few friends the certain Dri idea and they ALL love it as much as I do. It's around $7 at any drug store (rite aid, CVS, wallgrees .. ya know?) Seems a little pricey but a bottle lasts forever. And it may take a week or two to really start working so don't get discouraged!
The anitperspirant is in the green bottle, the "freshener" is in the bluish bottle, I suggest getting both. You put the green on at night before bed (it might burn if you've just shaved) and the blue one on in the morning.
It TOTALLY worked for me, I wish I had found the product years ago ...
good luck!
W.