Very interesting. I had never even thought about putting addressing with "Dear" or signing off my personal statement. I have recently submitted my application in, too.
Admissions commitee will know that it's a personal statement. So, I just started off like any other essay with an introduction.
I have seen a few personal statements end with a "Thank you for your consideration" or something of the sort, but I did not do this for my personal statement.
I guess if you are in a dilemma to put it in or not, I would first recommend that you have your entire personal statement written out to see if you even have characters left to include an address or an ending. I had problems at the end as I was running out of characters, and I still had more things I wanted to say!
I don't think that not including it or including it makes any sort of difference, so it's up to you whether or not to put it. But to me, it's not really a "letter"; it's a personal statement so I didn't use it.
Good luck with your personal statement! I can understand the frustration as I went through it myself for several days =/
