None of the above. It doesn't turn off at all, when the AC is on. At least for the 5 minutes I ran it just now.
So the compressor is running the whole time? The very end of the compressor, the end facing forward, is turning? And it stops turning if you turn the A/C off?
If so, that points to some different sort of issue. With not enough R134a in the system it will run a short period of time, until it sucks the low pressure side down too far and the low pressure switch shuts it off. Then the high pressure side will slowly bleed back into the low pressure side and the compressor will briefly turn on again. If it's staying on, that's not happening, the low side pressure is staying high enough.
If the expansion valve is stuck shut, it will move enough of the R134a from the low side to the high side that either the low pressure switch on the low side turns it off, or the over pressure switch on the high side will turn it off. Since it's staying on, that's not happening.
So if it runs constantly and doesn't make cool air, here are a few options:
1) Failed expansion valve. These have (afaik) variable expansion valves, which work better over a larger range of cooling. I know they can fail shut, I'm not sure if they can fail open, or at least so far open they don't meaningfully cool. But it's a possibility, the compressor is pumping R134a around, but it's not getting enough of a pressure drop at the expansion valve to evaporate/boil, so no cooling? Feel the two lines going into the firewall on the passenger side, there's a smaller one and a bigger one. Are they different temps?
2) The evaporator inside the HVAC system has leaves piled up on it and is completely blocking airflow. So it gets cold, but you don't get cold air, the fan's just blowing air around from elsewhere.
3) The compressor is just completely worn out/damaged internally and while it's turning, it's not actually making enough pressure to do anything. The R134a needs enough oil mixed into it to keep it lubricated, without enough oil it will chew itself up.
I think you need to get some pressure readings at this point. A cheap Autozone refill 'kit' will have a low pressure side gauge, that's at least some information on what's happening pressure wise. Check the pressure with the system off and 'at rest' (it hasn't been running for a while). Then start it up and turn on the A/C and see what the low side pressure does. Goes down? How much? Turn the engine off and see how quickly the low pressure goes back up (assuming it went down with it running).