Sounds like things are going very well between you and her. What I'd suggest is that you continue seeing her as often as you want to and see where things lead. You're enjoying the time you spend with her, so don't worry too much about putting a label on it and whether she can become your girlfriend or not. It may be that you aren't able to see her regularly enough to make this into a romantic or physical relationship, but so long as you're having fun that's so much less important. If you decide that you want to date someone that's closer to home, she can still be a good friend and someone you enjoy chatting with.
Also, don't worry too much about whether you're social enough. That's only going to take away from the fun you're already having. Clearly you're already doing a great job! One thing I've learned is that when you accept others for who they are, they will almost always accept you as well even if you have a few quirks
She may even find it charming. Keep your attention on what's going on and the people around you and let things flow naturally.