Just get your self a reliable razor, I'm sure if you just let your mom know she won't judge, shes been there, we've all been there at one point. No need to feel uneasy.
I don't understand why she wouldn't let you shave unless of course you're a male and shaivng your arms and legs as a male seems unnecessary. But I don't think you're a male since you mentioned you have this short dress to wear. lol So just go for it!!
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Explain her that you don't feel comfortable with your body, and tell her that all your friends started to do it (even if it's wrong ;)). I started shaving when I was about 12.