A buddy of mine came over and fixed my faucet. Thank god. The dripping was about to drive me over the brink. The problem was basically one of total decrepitude--the faucet was original to the apartment--but a quick trip to Home Depot and another cheapass faucet later, my dripping woes have ended.

Also, I sort of understand how a faucet is attached now. Not enough to risk installing one myself without at least an internet how-to guide, but enough that I'd probably be willing to tackle it solo with said guide, assuming I had someone I could call for help when the kitchen was under three feet of water and the cat was clinging to my scalp for safety.