After desperately using shampoo and suffering far worse for it, I decided to try applesauce. (A lot of people use that instead of baking soda, apparently.)
I tried it last night. My hair was super sticky again, no matter how much I scrubbed and rinsed. I had a light bulb moment: the problem wasn't the baking soda; it was my own sebum (oil) caked up in my hair. I tried spraying on ACV and combing through with an almost-fine-tooth comb, and I got some of the residue out, but my hair was still gross. I set off to do more research.
I discovered that hard water is the problem. It basically creates soap scum in the hair. (This isn't a problem with shampoo, which is a detergent, not a soap, and works with any water.) The top recommendation is boiling the water to soften it and mixing in the baking soda while the water is still hot.
This great post describes my problem exactly.
I was going to try this, but I discovered that boiled water takes forever to cool (but does make the BS fizz like nobody's business!), and I couldn't wait that long to wash my hair. I decided to try ACV in the shower, treating it like shampoo. I used water as hot as I could stand to wet my hair. Then I added full-strength ACV and scrubbed for a couple minutes. My hair started to feel slippery, the way it should feel during the BS stage, but not the acid stage. I figured I must be breaking up some of the residue and releasing BS that was stuck in my hair. I rinsed and rinsed.
Happy that my hair was in no way sticky, I stopped. My hair still felt slippery, and it left a slippery coating on my hands. I tried to rinse again with even hotter water. I got much of the slippery feel out, though it was still pretty slippery where I'm normally most oily (my temples and the back of my head). Satisfied that I did the best I could for now, I thought I'd see what it was like dry. It's still drying, and I can tell my hair's still questionable, though much better. I left my boiled-water concoctions in the shower and might try again tonight.
It's going to be a pain in the rear if I have to boil my water every time (or buy distilled water), but if my hair goes back to the way it was a few weeks ago, it'll still be WAY better than shampoo.