Is this a bad way to approach this situation with this girl?

Don't do the email, it could come off as creepy especially if she doesn't know who you are. I know how nerve wrecking it can be, but just go up and talk to her. I've always been somewhat shy as well, any girl I've ever had a thing with always approached me first. I started working out a lot and it truly helped me gain confidence. There was a girl in my university gym who was always getting hit on, I even had a class with her. Trust me, she was as close to a 10 as you can get. Way out of my league. One day I saw her at the gym and just thought "f*ck it". I went up and talked to her. Not even hitting on her, just randomly talked and introduced myself. I made sure to make her laugh, and honestly just acted goofy like I didn't care about embarrassing myself. It came off as confidence. I did the same thing perhaps twice more with her then one day I used the shittiest and most obvious excuse ever to get her number. Long story short, we've been dating a year

get over being shy and just talk to her