I am the same way and have been since I was 13 also. I even had the same issue of having to wear a uniform that left stains. You might have "hyperhidrosis" which is nothing to be afraid of or worried about. It just means you sweat a lot more than other people. I am now 29 and get botox in my arm pits (which is covered by insurance) and stops the sweating in the arm pits.
I would recommend not wearing heavy lotions or hair products since they can cause you to sweat even more. You can buy "armpit sweat pads" online and at stores and these absorb the sweat and are not visible to anyone. I also used to keep a second shirt just in case I needed to change (if it is the same one no one will notice you changed :-D). Thinking back to when I was 13 it was very difficult and I felt very self-conscious. When I mention how insecure I was about sweating to friends who knew me when I was 13 they all tell me they had no idea I sweat a lot so just try to remember everything feels 10x more embarrassing than it really is when you are 13.