I have used a wide variety of systems from ultralight backpacking to shower tent to public restroom to camper with full shower. I now have a shower in my camper that I use with a deck sprayer filled with warm water from my stove. It works great, but if water is short I will alternate with washcloth "baths".
Some of the best products that I have used are baby wipes,biodegradable cleaners, no rinse body wash/bath, microfiber towels and cloths.
Backpacking I have done full cleanups using a tiny bit of body wash on a synthetic cloth, with a full shampoo using a ziplock bag of warm water. Drying with a small microfiber towel.
The public restroom is one of the harder issues. Clean up and dressing in a stall is a balancing act. A quick shampoo can also be accomplished in the sink with a cup or cut off water bottle. Again a small microfiber towel.
When water and all else fails a little of no rinse body wash can do wonders. It is concentrated so you only mix a little bit, and it works great: https://www.amazon.com/gp/product/B0000 ... B16S8&th=1