See if there is a humdinger added, or a ct on the heater winding. A humding pot, with wiper to 0V is an easy mod.
You could temporarily add an extra capacitor to the screen supply to see if that is significant. Likely to need a new cap for the main B+ cap off the rectifier, as that cap gets the hardest life.
Do you have shorting input jacks or input plugs - if not then the 12AX7 grid may be floating.