Grounding does sound like the issue. Sounds like an extra ground current is flowing in the cable shield.
The power switch turns the unit completely off, no circuits have power. So there is nothing active in the pre to make a buzz.
When connecting ANY equipment with unbalanced lines, both power cords should be plugged into same outlet strip. Different outpets may create ground currents, which flow through the cable shield and insert noise in the signal path.
There also may be a difference in noise depending on exact position of the signal cables relative to the equipment. Move them around.
BUT, DON"T remove the grounding pin on the plug. It's there for a reason, which is safety. Leave it in place and solve the problem another way.
Unbalanced connections on any equipment are sensitive to externally induced hum and noise. Might come from a power transformer on the equipment, or some ground current in the building, etc.
Basically, that is why balanced lines were invented. But they are not the standard for typical instrument amps, so we have to deal with it.
System hum is often odd, even in simple systems. I have even seen hum induced from putting a preamp over a big power amp. The transformer in the power amp induced hum directly into the preamp circuits, again with or without power to the preamp.