if I unplug the antenna, it cuts off,
Your ground wire from the AVIC (black wire) is not correctly connected to the car's ground. The ground shield of the antenna wire is providing the power ground to the AVIC, and it was not designed to do that. You need to fix the ground wire problem. When you do, the AVIC will operate without the antenna plugged in. (Of course, there would be no radio reception.)
This doesn't help with your always on problem. The red wire to the AVIC has to be connected to a power source that switches off when you turn the key off. S4's comments are correct. In case it wasn't clear to you, the red wire must be connected to a different circuit than then yellow wire.