Now that everything is totally dry, I've put the Apple IIc back together. I'm really happy with how the handle turned out, less so by my skills protecting the "English" sticker on the spacebar.
Looking at the pinout in the Apple IIc Technical Reference, it looks like the internal 51⁄4 drive has the same pinout as the Disk II drives. I have two Disk II drives, both have the same symptoms when connected to the IIc.
Voltages from the DC-DC power supply all look very good on 5V, 12V and -12V while unloaded. UB13 also looks to be generating a good -5V (though it's not on the floppy ribbon cable).
Does anyone have an Apple IIc DC-DC power supply in working condition I can swap in to test?