Greetings!
I would NEVER use any type of public shower...
I have one in my van for quick showers, and if I want a long luxurious shower (about twice a week) I heat my 7 gallon hot water container, usually with solar, go down to any old park where I can have grass beside my van, toss up my 2 second shower tent, and have a nice long luxurious shower.
I've done this all over the country, even in very busy parks, and no-one has ever said a word to me other than people curious about my set-up. I don't think there's any laws anywhere prohibiting showering in a park as long as you're not committing indecent exposure or something.
Even when it's pretty cold it isn't that bad. I just dry myself off totally and put my warm clothes on before opening up my shower stall and exiting. While you're showering, the hot water keeps you plenty warm enough.
This thing is big too, 3.5' x 3.5' I think. More than big enough for me to take an umbrella chair in with me to put my clothes on and not have to worry about getting them wet. I take my 7 gallon water jug w/pump etc. inside with me too.
Just my solution...
Cheers!
The CamperVan_Man