There's a difference between powerful attraction, and the desperate need to be with someone. One just is, the other is a consciously or unconsciously manufactured state of neediness, usually based on your life experiences. You don't really sound like you're desperate. Maybe a little overly enthusiastic though....
Maybe dial it back a notch. That's where your enthusiasm can get mistaken for desperation. Maybe this is just my personal preference, but I don't think charging in like a bouncy ball, going "you wanna go out?!", getting rejected and going "Ok!" and bouncing out is super attractive. Maybe she digs that, maybe not. You basically want to be at a place where you can look her in the eyes, ask her out, and if you get rejected, maintain eye contact and smile in a way that says "I was ok before I asked you out, and I'm ok after being rejected too."
Recognize you're about to bounce out of her life either way soon. I've done the long distance relationship thing. It sucks, and I wouldn't recommend it. ESPECIALLY one where you haven't really gotten to know each other well before you poof. So I'd say go into it looking for fun and nothing else, and see if that's what she's about too. Cause if she wants a full-time boyfriend, yeah, maybe that won't work out so well.