Both of you are wrong.
Nah, I'm half-kidding here. If you don't do non-verbal, then don't try non-verbal. You don't have to worry about it, because if you try to fake it, it's going to end up just looking "wrong." Some of us can do that, but it's not easy to do. Stick with verbal communication. So the other two posters aren't entirely wrong here.
The important thing is not you talking to her, but just the opposite. Get her talking and keep her talking. Learn to listen. The only talking you need to do is whatever it takes to keep her going. People might get creeped out by how many questions I ask, but I don't care. I guarantee you are a lot more interesting than I am, so enough about me?I'd rather hear more about? Tell me what it was like when you actually met [famous_person_name_here] in person!
Hey, I know eye contact is weird. The other person might think she's talking to a brick wall. If you can't maintain eye contact, you need to communicate in some way that you really are interested in what she has to say.
I have this weird quirk where sometimes for me to actually understand what someone is saying, and I mean I REALLY want to absorb every word, I'll physically turn an ear to this person. If they get all weird about it, I'll just explain sometimes I miss stuff and I really want to hear what you have to say. It doesn't seem to bother people once I explain what I'm doing. You could do something similar?just be all embarrassed and awkward, apologize, and just explain that eye contact doesn't work for you but you STILL want to hear her talk about whatever. A nice girl is going to understand and go with it.
What's really funny is meeting people who have eye-contact issues. I've met people from other countries who absolutely will not look me in the eye. I tend to notice pretty quickly when that happens, so I just start talking to their shoulder if I even look at them at all.
What I'd do is give yourself to this person in very small, brief doses. Once they get used to the fact that you're just not an eye-contact person, they'll get comfortable around you and it won't be such an issue. Just give it time.