Pete, a house is controlled by a thermostat that has set points. Most of these set points are 1-3 degrees differential. Depending on the type and user interface that is needed.
When the hot set point is reached a 12 volt electrical current is sent to the furnace to turn on the fan. In return the Furnace sends a signal to the compressor and the freon runs through the evaporator thus giving you the cold air.
Now the compressor will NOT shut down until the low set point is reached on the thermostat.
Example: it is 72 degrees inside the house and you want it at 70. The thermostat has a 1 degree differential. This means the furnace turns on and compressor will run continuous until the desired temperature of 70 is reached. Once it is reached it cuts the signal, the compressor turns off and the furnace fan runs for about 45 seconds to help remove any frozen ice on the evaporator and make use of the already cold evaporator. Now when the thermostat hits 71 degrees it will kick back on. If you had a 2 degree differential then the inside temp would have to hit 72 before it would turn back on and so forth.
Now those 2 are very close in main operation but not so in makeup. Now in a car there is no set point. There is only high and low pressure. The compressor engages and starts to compress the fluid, once the high side solenoid PSI is reached the compressor is turned off. Now the pressure slowly falls until the low psi set point solenoid is reached. Then when the solenoid kicks on because the PSI is too low and the compressor starts climbing in PSI again to the high side, and over and over.
Now if the members compressor isn't kicking off and everything is good I would say the high side solenoid is not turning the compressor off. But again the only way to verify this and be entirely safe is to check the PSI readings on the High and Low side and see if you are in specs.
As far as your idea, I understand, however if you can't hold PSI at all then maybe the expansion valve is worn out allowing to much too fast to run through it so it can't hold the pressure, your low on freon, or the compressor is worn.
One last thing I would do,---->jbar<---- is take the harness off the compressor and make sure the clutch is working and not froze up. But again a PSI test is your best bet on figuring out what the issue is and the safest.