I think you got some bad information about the 3 month rule. I got my American passport on May 29, 2007 and made my first trip to Japan on June 28, 2007, less than 1 month after receiving my passport. I don't think you have anything to worry about.
As for expediting your passport, you can pay extra money at the post office when you apply for your passport to have it expedited. That's what I did. It cost me about an extra $100 ($60 for the processing fee and the rest for shipping charges). It takes about 2 weeks if you pay for expedited service.
Application Processing Times
I also had a roommate that drove to a regional passport agency and got his passport in a couple hours (granted this was back in the 1990s).
Regional Passport Agencies