Agreed about the follow up - keep it sweet and informative. You should have a bit of news to share. Also very much agreed on comments above the cover letter.
One thing that's really important to remember is, nobody cares at all that you are looking for an internship. The email you posted is polite, to the point - and utterly forgettable and irrelevant to studio owners.
Getting an audio degree is good (since this means you probably know how to turn on the equipment), but then you have to demonstrate that you are a do-er - more so than the other two dozen people who inquire about internships every week or so. So point out where you already have worked or what you already have engineered, i.e. "I recently mixed the full soundtrack for two web shorts and for a web-based videogame. Here are the YouTube links."
And if you want to get into post but haven't mixed any web shorts or games, the question becomes, why not? In that case you have to get on that, like, now. Do all your friends' video audio for free - vacations, silliness, no matter. Beg, borrow and steal sound fx libraries. Get cracking! Audio engineer is not a noun, it's a verb.
Good luck and have fun!