I would not camp on Mono Lake. Swarms of gnats hang around the lake. I have camped there once and never again. There are nice campgrounds on the road between Lee Vining and Tioga Pass. Given conditions this year, they will have mosquitoes. There are several free campgrounds near Deadman Summit (between Lee Vining and Mammoth) -check the Inyo Forest website. These sites are more high desert so maybe less buggy.

Showers at Twin Lakes used to cost $5. Seriously, have you thought of bringing a large-sized solar shower bag and put it on the hood of your car while you are out. You could rig up a shower curtain with an old sheet if worried about privacy. Public showers in the Lee Vining area are scarce. If you were to drive to Bishop, there are several places to shower and a public swimming pool. I bet there also is a public swim pool in Reno.