Switching 12V with Arduino Nano
Your circuit should work. I suspect that pin 8 is not set up as a real output (push-pull). That would explain the problems you are seeing.
To check this you can temporarily add a 1k pull up resistor from 3.3v to the drain when pin 8 is supposed to be low. If it doesn't stay really close to 0V then it would confirm it is set up as a open drain output and will not be able to drive pin 8 low.
You definitely don't want a resistor across the drain and source.
If you get it working right you won't need the other resistor either.