Luckily, I do have a donor system saved up for just this sort of thing, and thus, the testing commenceth! Unfortunately, it wasn’t as easy as just swapping floppy drives. While the donor system’s drive was compatible, the computer wouldn’t read it, despite it working in the donor.
The first thing to get tested was the floppy cable. It’s routed under the motherboard, so I had to get reasonably creative to test it.
It wasn’t the problem. The actual problem was that the moisture-damaged floppy drive somehow had damaged the motherboard. I replaced the motherboard in my computer (I forgot to take pictures of the process, sorry!) with the one from the donor machine, and then it was all fine. I may or may not try to fix the original motherboard, the thought of having the same everything as much as possible in this computer is quite appealing, I think.
Until then, drivers!