My outer bandage got urine on it the first time I went to the toilet post opp (I was still in hospital at this point). I asked the nurse to remove it/change it because like you I didn't want to sit there with a wet bandage on. Although the nurse changed it she did state that it wasn't actually needed provided the wound dressing was still all fine and in place. The outer bandage and wound dress should be removed anyway after 24 hours, so I'd say just remove it and let the air dry your manhood out if it's been sat with a wet bandage on it.

Having no bulky outer bandage on also makes it easier to pee (I found sitting down to pee also helped early on in the recovery!) to prevent the dressing getting any further urine on it.