Can moving wire location at circuit breaker reduce hum?

"Would swapping out the wires that go into the two separate rooms at the breaker make any difference?"

Give it a shot, you have nothing to lose!

Some people would prefer to understand the problem with say. DMM, rather than just throw “solution buccaci” at it, and hoping that they hit it..

One of my daughter’s friends is an electrician.  I’ll probably have him look into it.

Well done on the plan.

Maybe run an extension cord from the room with the good socket as a temporary measure?

Get several cheater Plugs and make sure the AC ground is not connected. These are male to female AC connectors that you can disable the AC ground. Many times hum is related to to a ground loop between units. Even if everything connected to the same AC outlet you can still have circulating ground currents.

If the OP already knows that the amps are humming without inputs, then it is not a ground loop.
And. The OP knows that a different circuit does not hum.

These clues are very good, and going against code with a potential for danger would lead me to believe that messing with grounds is likely be best to be avoided.

Knowing whether one side of the bus has a larger DC bias relative to the other would be a great place to start.
Without knowing why it is difficult to address the solution to the problem.
 

My point is just that if this is significant the OP should do everything possible to eliminate the root cause first.

So personally, I would reword ^that^ to”

The OP should do everything possible to determine the root cause first.