Reverse light stays on

I attached the wiring for this vehicle. The fact that you have replaced the switch, let's just check voltage in the location I showed while the vehicle is in park. If you have it at location one then your reverse switch is on or you have a short to power after the switch. Clearly unplugging the switch should cause that voltage at location number one to go away. If it does then the lights should turn off while in park. If they don't the we need to move to location 2 and check power.

Let me know what you find with this because either some odd is going on with the park/neutral switch or you have a short to power somewhere.

Nailed it. I would agree that the connector is the issue or the switch itself. I would look for a wiring connector kit and replace that instead of the entire harness. See the picture attached. Clearly this is not the connector you need but shows what you would replace. However, don't use those butt connectors. You will want to solder the wires and shrink wrap the wires. Those connections do not last.