Evan: Two things -- keep a daily check of the St. Andrews site. Late availability does happen on the Old Course, as that's what happened to my group last year.
Secondly, though September is a busy time, the ballot does give you a fighting chance. We entered the first time in 2003 and got the second tee time of the day (6:40 am, good god).
You enter your name on the ballot and can check via the Internet or phone by mid-afternoon the day before.
We were playing Carnoustie the day before our round at St. Andrews, and the gentleman who ran the hotel (Joe of the Carnoustie Links Hotel) entered us in the ballot and checked whether we had gotten in.
Anyway, depending on the time, it can be relatively easy to get through the ballot. Apparently July is a good time -- the GM at the Old Course hotel said all their guests got through the ballot in July 2003.
American tourism is still down, so you've got a good chance.
Best of luck and let me know if I can offer any advice. Allan Ferguson has a great book (
www.fergusongolf.com/book.htm) that will give you all the ins and outs of St. Andrews and many other Scottish courses as well.