Yea, you're sweating, so of course it is going to feel damp. I mean, you're working, so you *should* feel sweaty. Just make sure to change out of the damp shorts as soon as you get home.
+1 on what everyone else said. Many new riders choose shorts and/or saddles with way too much padding. More padding does not equal more comfort and can make chafing worse in many cases. Also, some shorts have more lycra or venting than others. Defer to the wise women at TE and get some opinions about what you should buy.
I know there's a few on TE who ride wearing underwear, but I can't imagine it. I don't wear it for any outdoor activity and when i went to a gym, I never wore it under my workout clothes, either. It just causes pain and bunches up. I had to break down and buy some Ibex boy shorts for warmth while x country skiing in temps below 25. Above that, it's just my nice, seamless long underwear under my tights.
Can you tell that I hate underwear?