First of all, Ram, it's nice that you asked her -- it sounds from her response that she was actually quite flattered (as everyone usually is, if they find you to be a nice person). It's a tiny bit of a cop-out to do it over email, of course, but it's still fine. From personal experience, I'll say that whether or not it's awkward later depends entirely on how you handle it. If the next time you see her you make a point of asking if you could talk to her for a minute and just say, in a somewhat humorous tone of voice, "By the way, hope you took my email just as a compliment, and that you won't now feel awkward around me -- for better or worse, I'm a man of action and always like to be direct about things," or something like this in your voice, all should be fine.

A friend of mine from home met someone in essentially this situation: he was interested, she was dating someone else, but they became friends and she ended up years later introducing him to the person he is currently dating. After all, if there is something you really like in a certain person she is likely to have friends with similar characteristics...

In short, I think to call something a long shot is to view the world in black and white in a way that is not exactly fair. You act in the world, you let dominos fall and things happen. Sometimes there are immediate positive effects, but always there is a certain opportunity presented. Perhaps you learn something about yourself, or meet someone new, or understand something about the world.

Hi Ram, Emily,

Ram, a while ago, at a time when I used to ask women out quite a bit, I ran into the following problem: Do you hit on a complete stranger?
For instance, I'm walking in a bookstore and see an extremely beautiful woman there - do I hit on her? Or in the airport, or on the street? (There was also a funny Seinfeld episode on this where he deliberates whether to hit on, I think, a tennis player.) After some deliberation I decided not to. My reason was the following: Essentially, the moment you cross that line once, every single woman you see is a possibility and that just seemed too much of a burden. Think of it, any woman you see on the street, in a bookstore, in a restaurant, each of them is a possibility. It's quite a difficult life, no? A certain limiting of the possibilities made life a lot easier.

But, life as a student is simpler. You constantly meet women, and though you always welcome more choice, limiting to university life still leaves you with quite a bit of work. At least in big universities.
In case you are after university, then the obvious limiting lines become very restrictive and maybe you'd just choose to live without any line, accepting the burden which comes with it.

Besides that - the long shot? Who isn't a long shot? Look at it from a larger scale. If you're looking for a one night stand, the long shot has meaning, but If you're looking for someone for a very long term relationship, what does it matter if she turns out not to work out before the first date, after it, after a month or a year. As you're still dating most of your dates were long shots who misfired, so what does it matter if it's a long shot before or after dating. 
If she would have accepted and you had a couple of awkward dates would that have been better? Perhaps ego wise, but come on. Everybody is a long shot if you look at it in the long run.

And why do you feel awkward? You asked, she has a boyfried, where's the harm? It's not like you did anything wrong.
Or do you feel something like this policeman?:

Great video and context, couldn't help but laughing out loud...