A great trick to getting over shyness (I was once extremely shy) is to ask questions about the other person. It's helpful to think of a few in advance so you can avoid questions that get one-word answers. Some can include things like:
"If you won the lottery tonight, what would be the first thing you would do? What would you do tomorrow?"
"If you could meet anyone famous from the past or in the present, who would that be and what would you say?"
"Name one thing you're dying to tell someone that you never told anyone before."
And since you don't want to become her best friend, but rather a boyfriend, (and she acts like she's enjoying the Q&A) you can also throw in a few questions like, "how do you like to be kissed?" "what do you like most in traits in guys?"
Since you feel you haven't been too talkative, you can preface the hanging out time together by saying since we can't always find something fun to do, I thought it would be fun to play 10 Questions - so I'll start by asking you some and you ask me some.
I dated a guy I thoroughly liked just after high school. We rode around in his awesome 1964 Mustang (which I loved), and every single time I'd drift asleep with my head on his shoulder. We really liked each other a lot, but I was SO relaxed that I never stayed awake! If he had been a bit spicier and direct in his approach we would have continued dating. Instead, we amicably broke up and he dated a wilder girl (probably good for him because he was so reserved) and they got married! So give this a shot, and if it doesn't work out, keep in mind that a better match for you might be around the corner (literally!).