My sink faucet continues to drip even after the water valves to the sink have been shut off. Help, please!?

It's a two-handled sink faucet. It's been leaking for about a month now. It's a very slow drip, but still a problem. I've already examined and cleaned the spout of the faucet. Even after I shut off the hot and cold water valves to the sink, the water still continues to drip. Turning the handles to drain the pipes of leftover water and shutting them off again, the faucet still continues to drip. I do not want to have to pay a plumber to fix this problem, but I have no idea what the problem could be. Help, please!