Use my road bike and buy a trailer for a tour, or buy a real touring bike?

I'd like to ride from Seattle to Los Angeles this spring and don't know whether my current bike would work. I have a stock 2006 Trek 1600 that I bought before I knew a lot about bikes. It is nice and light, but the riding position is very aggressive and would not be comfortable for a series of rides. I've thought about buying a Bob Trailer and attaching it to the back.

I could also buy an entry level road bike for about $900 plus money for panniers, shoes, etc.

Do you think the Trek 1600 would work with a trailer or should I spend more money for a real touring bike?