Washing them out in a sink is adequate. Any soap works OK. Camp suds, any shampoo, or even bar soap all are fine. I usually start a tour with a few ounces of baby shampoo and replenish with whatever I find. The little bottles of body wash or shampoo are usually available for free if you are moteling it. Which ever soap I have, I use it for clothes, body, and dishes.

I take advantage of coin operated washing machines and driers if they are available, but don't go out of my way to find them.

Personally I find 1 or 2 pairs of shorts and 1 jersey along with 1 set of off bike clothes to be enough. FWIW, I don't get the bib thing.