Should I tell my teachers about wanting to change my name?

Me, my friends, and my mom have agreed on a name that we think suits me, and a teacher has asked before if I wanted to go by something else (I look pretty gay lmao) But I'm not sure if I should actually tell people that I wanna go by that name, especially my school

My school's pretty conservative, and is very into gender roles and religion and stuff so idk how they'd react or if they'd be more strict with me or anything

Also I have extreme anxiety whenever I talk to someone who isn't a close friend, and idk how to bring up name changing to my teachers

Any suggestions would be nice :))