Regardless of how tightly I turn the faucet off, there is water that flows out. When I attach a hose with a spray nozzle, the water finds a way to leak out either at the faucet where the hose is connected or around the spray nozzle. There are 4 other outside spigots that are not having the same problem.
Oops, I think I misunderstood the question.
It sounds like this is a "frostproof" spigot. If so, there should be a small white plastic cap on the top of the spigot just behind the knob. A picture would ensure that a proper identification is made.
Most of these spigots can be rebuilt. However, it may be more efficient (time wise) to replace it (depending on brand, age, availability of suitable parts). But... this depends on how accessible the body of the valve is in your structure. Again, a picture is going to help a lot.