Can't they use Botox under your arms to keep you from sweating? I have heard that is helpful for some people. Might be worth asking your dermatologist about.
As far as removing smells from cotton, have you tried color-safe bleach? (Clorox 2 or similar) That usually helps. Or, if you know someone with a steam washer, wash the shirts using steam in the Sanitize cycle. I have a Samsung washer with steam, and they get my husband's disgustingly stinky rugby socks clean and smelling nice, and I just use regular laundry detergent - so I would say that the sanitize/steam cycle should do the trick!