As you've probably guessed, it's time for a new T&P valve. It's a pretty cheap and easy fix, but considering how long the valve has been in there it might take a bit of force to break it loose.
It's a good thing you mentioned the leak exists with the heat off. If the drip only happens when hot, there could be a problem with the thermostat allowing the water to overheat and the increased pressure and temperature can cause the valve to unseat like it's supposed to. That's also why it's very important to never cap off or disable the T&P valve as it is a very necessary safety feature.
While your at it, have you checked your anode rod lately?