I've had this issues since I was a teenager. When I was in high school, my mom took me to a dermatologist, and he prescribed a prescription-strength antiperspirant that I had to put on at night, wrap in plastic wrap - and it tingled and stung all night long. And it worked, but only temporarily.
Now, you can dry over-the-counter antiperspirant, it's called Certain Dry. You apply it at night (you don't have to wrap it in plastic, thank goodness), and it lasts a week or so.
The only permanent fix for this problem is surgery or botox injections in your armpits - and I'm not keen on either. I just use the Certain Dry on occasion and regular deodorant/antiperspirant daily. And just a hint of good news, this condition does get better as you age. I've lived with this for almost 30 years, and it's definitely not as bad as it was in my teens and twenties.