The pots are probably fine. There is no need to replace them unless they are scratchy, loose, poorly tapered, or the wrong value for your preference.
If a pot does not fully mute the signal when turned all the way down, it is almost always a grounding problem. Unless the third terminal has broken off of the end of the resistive element, the pot reduces the resistance from the wiper terminal, to that terminal, to within a couple of Ohms of zero. This means that if the third terminal is grounded properly, the signal cannot flow to the output, as it is almost directly shunted to ground.
When there is no ground, you get similar behavior to what you have described, because the pot is only providing a series resistance from pickup to output. The resistance of the pot is not high enough to completely stop the signal from the pickups from reaching the output.