OK, me too, I was active in high school sports and sweat like a horse under my arms. Yes, there was no b.o. but a lot of stain from the salt that leaves your body. So, like you I tried all kinds of antiperspirants and only wound up with skin rash and more shirt stains. Then a little old lady told me in the good old days they didn't have deodorant, they had baby powder, baby talcum and corn starch.
Yep, I tied baby powder, no deodorant as it gets sticky under there, it worked a while but during sports it washes away and you have to have a small tub handy, also I used baby powder and talcum powder it worked and kind of smelled like a baby but hey, the girls liked it, last I tried corn starch ( not from the box) use baby powder with corn starch, it worked wonders and I just spray a little cologne or body spray (for men of course) and wallah, I'm dry no stains and no smelling sticky mess like deodorant.
Another Place, Another Time