Hey there-
So I can only speak from my experience....I wasn't thinking at all about my junk--all I could concentrate on was having my boobs removed. However, ALL of my dysphoria was chest dysphoria, I have no lower dysphoria, so my experience differs from yours in that way.
As far as being awake enough to get dressed after: No, not immediately. I'd worn boxers into surgery. It takes a bit of time to get fully awake (Heck, I wasn't fully awake when I left the hospital; I was craving coffee after surgery-my partner drove us to (a new) Starbucks on the way home. A couple months later we drove past that Starbucks and I said, "oh we haven't been to that new Starbucks yet!" I totally didn't remember going there after surgery! The memory did come back though).
Anyway, I was covered under a blanket from the time I woke up from surgery til I left (I was freezing after surgery and I think that's pretty typical), so no one is going to be looking at your crotch.
Maybe that was helpful? lol