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You seem like a pretty smart guy, and "day game" works towards the smarter guy and against the "dumber, stereotypical frat guy."
Hey now.. I'm a frat guy

no where near stupid either if I do say so haha and yes it is called day game.
anyways besides the point. You just have a severe case of Approach Anxiety and I'm not going to speak for everyone but I know I did too. You just need to get over it and do it. "You miss 100% of the shots you don't take" so you're already counting yourself out with her.
You say you get a lot of looks, that's a good thing. Don't be self-conscious about it. It's a good thing.
As for what to say, you just need to find what style fits you. Just get a pretty simple routine or opener. Don't rehearse it too much, however, or you'll just think too much. Go up with
confidence and opener her. Then show her you're not just the typical guy who's she's used to talking to. There are plenty of post on here about what to say. Just read up.
Go out and do it. Don't sit and think about it and let the butterflies twitterfuck you. The worst (hopefully

) thing that can happen is she's not interested in you and that's not terrible since you can always learn from your mistakes.
Get out there man!