Hi there lannyman8,
Unless you've got a hot date I wouldn't worry overmuch about "proper" washing for a day or two at your water-free camp. Perhaps the first thing is to consider the effort you put into your hiking - you say you'll be doing a lot. I like to hike with my body at the cool side of comfortable and adjust clothing and energy input accordingly. This also reduces your hydration demands as you're sweating less, and consequently sweating less into your clothing. Also I use merino wool base layer as these can be worn for several days without getting too skanky, unlike some man-made base layers which can last barely a day.
I hike a lot in bear country and consequently go deodorant, toothpaste and soap-free for several days at a time. (I believe bears like to eat nice-smelling people.) I find as long as I attend to my teeth (with water only), and after a hike my head, face, armpits, crotch and feet with a flannel soaked in a small amount of water I can feel good for some days. I think Toddy suggests two flannels, one for top and one for tail - colour code them if you go this route. I also occasionally use unscented wet wipes. If you use soap you need water to rinse it off.
The most important thing is hygiene during food prep and consumption and I use alchohol gel for this. Bombay bum in the boonies is not something to be countenanced.
One final thing, IME if you're hiking with a number of people over several days you'll probably all smell the same after a couple of days and the "clean" one will be ridiculed!
Have a great trip!
