If you listen intently, and/or feel on counter tops, you should be able to detect if the compressor is coming on. If in doubt have someone start the AC on "FAN" only then switch to "Cool" with the thermostat set appropriately to get cooling. Again, you should be able to detect if the compressor is running. Also if you have an electrical power monitoring system with a display, the AC with compressor running should pull about 12 amps (or more). Cooled air from the vents should be 20 degrees or more cooler than air measured at the intake.
The compressor does have an overtemperature switch mounted to its top that opens when it gets too hot from ambient conditions or cooling system problems. I was able to extend the life of my unit a few months by hanging a room fan on the shroud aiming in at the compressor. I eventually had to replace the top unit.
Top units are available on E-Bay, but you have to get this heavy piece of equipment on to the roof. Otherwise replacement isn't a giant job. Power OFF, Remove trim inside, remove 4 bolts inside going up, on the roof remove the shroud, unplug the 1 electrical connector from the bottom, lift off old unit, remove old bottom gasket on roof, prep roof for new gasket (normally included with new top unit) put roof unit in place, plug in electric plug, work with someone inside to get 4 bolts started (you will probably have to move top unit slightly to get it aligned right). Tighten bolts hand tight (too tight or too loose cause condensate water leaks inside through the gasket), test unit, and replace inside trim and shroud. Then get the heavy old unit off the roof and to the dump. You might want to retest the tightness on those bolts after a few months.