My girlfriend and I met when I was black out drunk, how should I handle this?

I went home with her that night. She wouldn't give it up that night but we did other things. When I came to in the morning I realized she was really cool! Of course, normally it would be hard for me to begin a relationship with a girl like this because how would I know she doesn't do that with other guys. But now I really trust her and want to be with her. She is the first girl I've seriously talked too in years maybe ever. She said I am her first in five years. I'm 23 and she's 21. What should I do?