Where are you from? I'm sure culture shock could be part of it....
I dont blame you for not wanting to seek out a counselor from the school. It IS a rather embarrassing topic and from the sounds of it, your school is really tough and strict.
Ultimately you need to figure out how to relax in a classroom setting. If that means asking for a standing hall pass then so be it. I used to have anxiety over going to the bathroom in public because i was in school and some poor girl was going poo and you could hear it and some idiot girl started laughing and I was so embarrassed for the other girl, it traumatized me. After that, i would have a lot of anxiety about going to the bathroom in public facility or even being in public and having to go. I eventually got over it but it was a literal and figurative pain in the rear when it was a problem. Just try to figure out how to help yourself relax. Honestly, if I was having a "go or die" moment, i'd just leave. All my teachers would always say "if it's an emergency, just go." So there's always that option. Just try to figure out something that works for you.