Since the buzzing occurs even when the engine is not running, that would suggest it's electrical in nature. A solenoid that is either getting low power, has an intermittent connection, or is receiving a pulsed signal could/would buzz similar to how you describe.
Since the only changes are the installation of the used transmission and the repaired wire, I think you are on the right track to focus your efforts there, but I would first confirm your battery is charged and you have good system voltage. Assuming the system voltage is good, then track down a wiring diagram for your car's transmission to get a better understanding of what the red wire powers along with what the other wires power to help you narrow down which solenoid is making the noise so you can further focus your efforts.
-Rod