It is strange how often my combo lock doesnt seem to open for me there either
LOL. Works for me too but in my case it is also because my glasses are inside and the numbers on the lock are so small. Whatever works i get to be naked :-)
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.