I do my laundry by hand, in the bathtub, or in the sink. You are going to need a quick dry microfiber towel in order to dry faster.
I don't know if you know how to dry. In case you don't, you need to wring out the clothes as much as you can, lay them flat on the towel, roll it up, like sushi, then step on the towel or wring them in while in the towel. Then you can hang them to dry.
I've been without a "real home" for three years.
The most important things to me:
Those are pretty much the MAIN things for me.
There's a really great book I would HIGHLY recommend you look at. It's called the Modern Nomad's Backpack. Look, I was raised as a nomad and I learned things I had never thought of. This lady literally thought of everything down to a collapsible bucket. It would really help you.