By Thursday, the water wasn't receding at all. I resorted to scooping the water I could out of the sink with a big mason jar into a bucket and dumping that into the backyard.
I disconnected the pipes under the sink to get the rest of the water out, and tried to snake the drain with a little manual snake from Amazon. The thing kinked up and turned on itself (behold my DIY skills) so after an hour of that, I did what I hate doing: admitting I can't fix something myself and calling a professional.