So ladies - I went ahead and tried it. I purchased ORS Herbal Cleanse and dry shampooed my hair. This stuff is basically just alcohol with a few essential oils (lavender and basil). The consistency is similar to hand sanitizer (it's basically a gel in the bottle, but when applied, it turns into a liquid). I followed the instructions and applied a generous amount to my scalp, let it sit for a little over a minute, and then took a warm, damp cloth and wiped it off of my scalp.
It was nice to feel like I was cleaning my scalp, but I definitely miss running water through my hair and cleaning my hair and scalp by lathering shampoo. I didn't add any oil to my scalp right away because I like to let my scalp breathe for a bit before slathering stuff on it again. BUT - my scalp itched quite a bit after I used this. I'm not sure if it's because it dried my scalp out a little, but I definitely had heightened itching.
In addition to cleaning your scalp, you can also apply a generous amount of this to your actual braids, cover your head with a warm, damp towel for a few minutes, and then wipe your braids off to remove the residue. I would like to try this, but because the primary ingredient in the cleanse is alcohol, I'm not sure if it will be a bit too drying for my hair.
Overall, I think I will use this again in order to avoid washing my braids with water, and hopefully minimize the potential for tangling.