Shave: Grow a beard, truck stop restrooms in the morning, you'll blend right in with the truckers doing the same.
Shampoo: In the city swimming pools are a decent bet, sometimes you can just ask if you can just shower and they'll let you. Gym memberships like Planet Fitness can be reasonably priced for just the shower. In more remote areas, campgrounds are a good source of showers, sometimes free, sometimes a modest fee. Plenty of van dwellers will even setup a cleaning area in their van, usually some sort of bucket they can stand in a pour heated water over them selves, or at least a sponge bath.
I'm not a fulltime van dweller, I started off just skipping out on hotels during 2-3week long road trips, eventually moved into my car (Pontiac G6) for 6months in the Boston area. Now its more sporadic, a month or two here and there while doing some long distance hiking.