No, I don't wear undies. I'd venture to say that most of us don't. The conventional thinking is that underwear is liable to bunch up or otherwise chafe you when worn underneath cycling shorts.
Have you tried wearing underwear underneath your cycling shorts on your new bike? I'd venture a guess is that your discomfort has more to do with the stock saddle (assuming you haven't changed it out) that came on your Trek and/or your shorts themselves. Many of us end up having to change saddles to find one that works for our particular anatomy. It pays, too, to invest in quality cycling shorts. You don't need to blow your entire budget on them, but in my experience, there is a difference between a cheap pair of shorts and higher end shorts in the quality of the lycra and of the chamois. TE has a wonderful selection, and if you need help choosing, just give them a call. They've road tested many of them.
As for the dampness, it's never really bothered me. Again, a good quality chamois should be absorbant. I use chamois cream on most rides, so my shorts get kind of, uh, goopy anyway.
While I don't want to sit around in them for very long after a ride, while I'm on the bike, it don't notice it.
Live with intention. Walk to the edge. Listen hard. Practice wellness. Play with abandon. Laugh. Choose with no regret. Continue to learn. Appreciate your friends. Do what you love. Live as if this is all there is.
--Mary Anne Radmacher