If you can find a public swimming pool in town, every one that I know of provides showers.

I and a couple of other people did a two-week trip in 1988 through the Sierra together. Total camping fees & a couple of motel stays came to $70 a head. These were all developed campgrounds with showers, though some places showers cost extra, as in Yosemite.

If you have enough BTU power, you can heat up a quart or two of water to boiling and mix it with cool water in a gallon jug or even big water bottles, then use those to take a shower. Bike shorts stay on if I'm in public...