Walk to the urinals and take a piss.
Weigh yourself.
Bring toothbrush and toothpaste and brush your teeth.
Trim and file your nails.
Apply body lotion.
I belonged to a men's athletic club back in the 80s. Everything you describe and what I've added were common practice at the time. No one bother wrapping up in a towel. Getting dressed was the last thing anyone did before leaving. Showered in an open bay gang shower. The urinal was out where it could be clearly seen. The only thing private there was the toilet -- not sure why that was an issue.
Didn't have cell phones back then, but you could hold your phone inside your locker (so no one thinks you taking pictures of them) and check you messages.
Well, I always use a medium sized microfiber towel to dry off with. Its nowhere near large enough to wrap around or cover myself with. I always dry off in the shower stall (no curtain or curtain open) and walk back to my locker with my towel in my hand. I use a combo lock so it takes time to open and sometimes I have to try a few times. You could also, with a small/medium towel, use a key lock and lose the key down the shower drain be fore you walk back to your locker nude. Then ask someone for help or ask them to get the front counter person for you, meanwhile standing there completely nude while you wait, which might take a while. I might have to try that when I visit a different gym!



