The shower is fixed! In the end, they had to replace the faucet and the pipe that the showerhead connects to, and it cost a hundred and twenty bucks*, but I now have a working shower!
Thanks, Auchinachie** Plumbing! Your plumbers are awesome.
*Welcome to homeownership. Get out your checkbook.
**Pronounced Ah-HAWN-ah-hee. Yeah, I don't get it either.