"Is it normal for the outside of the drain pipe to be wet."

Yes,it is normal. If the condensate is dripping onto something that might be harmed,any hvac tech,handyman or average diyer can cure the drip by insulating pipe. If it isn't dripping on things it shouldn't,don't worry about it.

klem1 thanks for your response. My husband is going to insulate the pipe. We just didn't know if we could do that.

You are welcome. Quickest way is foam tube you can buy at hardware and big box. Tape the slit in tube. Any foam will work but that which come's in a roll is better than sticks.

Thanks klem1. I'll tell my husband.

Foam tube is good for straight runs only. Get foam tape for everything else.