...augur didn't work. Which is to say, I was able to run it all the way to the next cleanout (turns out there's another one after the pipe turns 90°) and then run it from there all the way to the next 90° bend.
It still takes about the same amount of time for the sink to back up. *sigh*
At this point I'm out of ideas. I know the clog is still there, and I know it's upstream of the second cleanout and downstream of the first. But I cannot feel any obstruction with the augur between them, at least nothing serious.
Next step: blow the bitch out. I'll get one of those "water weenie" things, like this, and try power-flushing the pipe clear. The blockage is obviously nothing solid, so hopefully water pressure will do the job.
Of course, if I had simply listened to the little voice that said "buy the water weenie too!" I'd already be doing that. But it's too late to get one tonight, so I'll get one tomorrow and give it a go.
Meanwhile, after cleaning up downstairs, I stripped naked and put everything I'd worn (two shirts, two pairs of sweatpants, everything) into the washer, and I came upstairs naked.
Mrs. Fungus: "Well, aren't you fancy!"
...immediately got into the shower and hosed myself clean. So at least I no longer smell like drain-pipe grease. I've got that much going for me!
The nice thing is, though, that the pipe in question is 1.5" PVC. If I had to, I could cut it out and put in some new stuff and be none the worse for wear. I don't want to do that because it's a pain in the ass, but I could if I had to. Nice to know that I have some options here if worse comes to worst.
Hopefully the "water weenie" method will do it, though.