I'll throw in a thought here as well...
The sort of high frequency hash that you're getting is often the result of a feedback or gain problem that's outside the audible sound spectrum. You can't hear a 30KHz tone, but the opamp in your EQ circuit faithfully tries to reproduce it and may be driven into overdrive by something you can't hear.
When this happens, it's often because something is out of spec in the circuit and is causing feedback at some superhigh frequency.
One cause of this is faulty high-resistance connections. The previous advice hits a lot of the points to try here - turn the pots, exercise the jack and pickup switch. Over the years, a film can cover the contacts which will cause it to act like a resistor instead of a wire, and that resistance in the circuit can start the out of band feedback problems.
It is also a good idea to disconnect, clean, and reconnect the push-on molex connectors that connect the pickups and jack to the EQ boards. It will probably be sufficient to pull the connector and see if the pin that it's stuck on is shiny. If not, then use some contact cleaner or a polishing cloth to clean the pin surface. You might try something like Cain DeOxIt which is a red colored contact cleaner for this purpose. I don't know if I believe that DeOxIt is as magic as the manufacturer would have you believe, but you can buy it in a little pen with a fiber tip which actually does an excellent job cleaning small areas like this.
The other source of this sort of problem isn't as easy to deal with, as it may be a failure of a component on the EQ board. It's unlikely that a resistor or cap has gone bad in here (they are the usual suspects in high-power applications like an amp, but unlikely in a low power situation like this). One of the opamps may have failed through which would probably require a trip back to the factory to diagnose and repair.
You can think of the electronics in the bass as a chain of blocks, first the pickups and humcanceller, then the filters (which is also the first amplifcation stage), and finally a mixer to add the two signals together. Since the humcanceller trims aren't having any effect, then the pickups and humcanceller circuit are probably OK, and since the noise is coming in before the filters, then it's probably the early gain stage that has the problem.
Some stuff to try, anyway. Good luck
David Fung