Am I starting to annoy her? Am I being too eager/pushy?

Well I am not 100% positive but it seems like she's playing you. I don't think that she wants to meet up in real life otherwise it would happen already. But that's just my opinion. And it's not a matter of being shy but one has to be careful when meeting someone on internet as you want to make sure you're not getting in touch with some freak. Anyway you're not annoying her. Just ask her directly. Yes or NO? that's it. Good luck.