It is possible that the new computer has one of the types of power supplies, where, when the computer is turned off, there is still a little bit of electronics that are "live" that are waiting for your "signal to power on" via the soft power button.

Some history:

Old computers had actual full giant AC switches which fully cut off the mains by actually interrupting the main power wires directly. Throw the switch and there is literally no AC power reaching it. Actual full physical switches where the main AC power goes through the switch.

New computers have power supplies which are driven off of a small temporary-contact soft button which controls a piece of firmware that interprets how long you pressed the switch. A short press will send a software command to the PC that can do things like tell the computer to hibernate or wake up from hibernate on a short press, whereas a long press will tell it to actually shut down the power.

All of the above requires that the power supply still needs to draw a bit of power, even when it's turned off. There's a small bit of circuitry that interprets those button presses.

That's probably why the "controller" AC plug is still staying on. It's sensing that your PC is still "on" to a very small degree.

Question:

When I've seen those "master/slave" AC plugs (I use one in my electric guitar rack in fact), there's usually a sensitivity adjustment somewhere, a small recessed potentiometer that you adjust with a small flathead screwdriver. Does your have that? Maybe you just need to adjust it upwards.