Of course I ask the internet what to do with a broken garbage disposal. Much to my surprise it actually helped! For one thing, the internet told me that garbage disposals typically have a reset button on the bottom. Pushing that made it go back to making that useless hmmmmmmmmmm noise at least.
I poked at the gross milky chunky nasty contents with a wooden spoon hoping that maybe that would help and it didn't. Then I tried putting a little dish soap in and miraculously it worked like a charm. Even without the garbage disposal on it suddenly started draining, and then when I tried the disposal it sprang back to life, making a more proper grnkgrnkgrnkgrnk sort of noise.
So yeah. Palmolive dish soap: the poor man's Drano.