If there were an effective way to dry-wash hair, I'd be much more likely to camp at campgrounds without showers. Perhaps I'll try one of the products in that regard that they sell at REI.

If things were so bad that hair washing was not doable then I'd be likely to at least cut my hair very, very short. Shaving it is a very reasonable alternative. I have lots of hats.

Or I could just use my dog's waterless shampoo. These products are popular at dog shows.