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.