If the contact cleaner does nothing, then I'd say that volume control behavior suggests it's capacitive coupling, or at least high impedance coupling, to a grid. It would most likely be the 12AU7 grid, since that is the most sensitive part of the circuit.
One possibility is a tube whose heater voltage leaks to the grid. Trying a replacement tube is the easiest way to test for this.
Another possibility is something off in the layout, with a heater power wire too close to a grid terminal. Perhaps you could post a picture of the wiring? You may have to unscrew and push aside the PC boards to get a good shot of the tube socket wiring.