Email is just fine given the time constraint. It worked for me. As far as the addressee, you could probably just email email@example.com, or firstname.lastname@example.org, if you get my drift.
Phone is always faster and better than e-mail. Period. End of story. Yes, I am sure e-mail "worked", but you don't know how much you would have received had you made a call as well, do you? The world is still a high-contact environment.
This won't go over well, but come on, we are talking about thousands of dollars and you think you will do just as well by e-mail? I'd like a lawyer who is willing to put in at least a phone call on my behalf if thousands of dollars are on the line. If they don't do it for themselves, why the hell will they do it for me?
...So when the finaid office intern answers, they can listen to your whole story and do nothing for you. Writing the email is like writing an additional personal statement, and you send it right to the one who matters. If they accepted you, they want you... They'll read your email, and pass it on to others who matter.
Then call to make sure they got it.