When you say, running, do you specifically mean the compressor is running or the fan. Does the fan always run to move air in the cabin and you are saying that the compressor never comes back on?
Often the temp sensor is in the return air duct where you can see it, it looks like a copper wire with a fat end. pencil size usually. This needs to be in the air flow, not tucked away. Unlikely the issue though in your case.
All the wall thermostat does is operate like an on/off switch. The temps control the switch. You should be able to literally pull the wall wire out and connect the two wires together and this will turn the compressor on. If you suspect a bad thermostat you can try this, but you will have to manually un-do the wires when temps are reached. Not very good for normal use, but good for a diagnosis.
Voltage or freeze sensor of course more probably causes....