There's no auxillary 12V power. The 12V is completely turned off when the computer is shut down or put into standby, but the 5V standby supply, on the violet wire of the main power connector, continues to operate, unless the AC is disconnected. Get a power strip or battery backup, and shut it off to stop the coil whine.
Coil whine can be from loose windings or from failing capacitors changing the duty cycle of an oscillator or causing the driving transistors to switch less abruptly. Capacitor failure is not always visible.