OK, very interesting. Failure of bypass/filter capacitors can also lead to instability. When replacing those caps, correct polarity is vital. Oscillation about a setpoint can be caused by excess gain and positive feedback. Also by not enough derivative feedback.
So it is possible that capacitors have aged and lost value. Or a noisy pressure transducer. That problem is often called servo hunting.

 

All of the PID settings are set by an external controller which is configured by the end user or undergoes some auto tune function programmed into the controller. So if they run an auto tune on working parts this eliminates the extreme oscillation but on the part in question that function does not have the same effect.

Since filter cap polarity is vital, and there's only one polarized capacitor on this board is it safe to say that would be the filter cap?

 

Now the mention of "HUM" from the solenoid valve coils, a question as to if that hum sounds like the mains frequency, or is it much higher??
AND a question as to if the valves on controllers of other machines hum.
an unintended oscillation in the solenoid driver portion of the controller could certainly lead to disturbances in other portions of the controller system.
AND a check for any AC presence on the DC power supply feeds might be revealing. I am guessing that the supplies may be for analog, digital, and valve driver portions on that PCB. That would be a reasonable scheme for a complex system, such as this one.

 

personally i would look for identifying marks like manufacturer, product number etc. (not for PCB itself but for the assembled unit...). next step would be to track down user manual... possibly also service manual.

 

Now, a day or more later, I recall that for forcing stability in a servo there may be an actual dead-band, an area between demanding adjust in one direction versus adjust in the opposite direction. That would amount to a very small area of error response with lower gain. THAT value may not be quite right.
Aside from that, there could easily be mechanical issues with the two solenoid valves, either leakage or hysteresis. And with the valve being controlled, there could be non-linear friction, also known as stick/slip.