First series of tweaks to my MSSH.
Connected the 0V rail to earth with a 100Ω resistor. This eliminated the noise picked from the floating inputs.
Doubled the value of R13 hoping to lower the back ground noise, it didn't change anything.
Added an identical resistor in parallel with the one in the CRC filter, because the original 7Ω dropped too many volts and overheated. I thought it would make the noise louder, but it didn't.
Connected the volume pot's metal casing to ground. This eliminated the static noise when touching the pot.