Getting things apart was not an issue, and once I had removed the sticky tangles of hair and wiped out some of the foulest smelling sludge I ever care to come in contact with, I was ready to put everything back together. That's when I found the problem.
The plastic nut on the drain pipe was split.This is the nut that compresses the rubber seals between the plastic drain pipe and the ceramic sink bowl. Without a good tight seal, no doubt we would end up with puddles under the sink, and there's no way the cracked nut would be able tot do the job.
Now they say laziness is the mother of all invention, and my laziness was not the least bit interested in driving back across town. Fortunately, I always have an assortment of zip ties on hand, and in a fit of indolent genius I found the answer.
One hefty zip tie later and I could tighten the plastic nut down securely without the crack opening up and skipping down the threads.
I doubt the fix will last forever, but I can't pass up the chance to brag about such a nasty hack.