For what it's worth, bb, I wouldn't try to cover up your social awkwardness. I mean, don't turn it up to 11 or anything. But it's who you are. Own that shit. You can acknowledge that it's something you work on/struggle with and at the same time let it be an endearing quality. Besides, believe me when I say that she already knows that you're socially awkward. If she doesn't, that means she's adorable or her own social issues match yours. And if she's still friendly, then asking her to coffee is a great way to start exploring the depth of that friendliness. She'll either turn you down, or she won't. But then you'll know.
But do NOT do the whole friendzone thing! You've interacted with her in a group setting, she knows who you are, and she's friendly. There you go, dude. That's enough groundwork to ask her on a date right there. Pursuing the friendzone nonsense is going to make you a friend. And if that goes on too long, one day she'll excitedly come tell you about this new guy that just asked her out! Isn't that great? NO! Instead, YOU go be that new guy. Carpe Diem, man!