Next time it , the a/c, shuts off, push the Defrost button and see if it energizes the compressor. Eliminate the push button assy. to take it out of the mix. Also, does the whole climate system shutdown or just the compressor.
Too much freon, gets hot, shuts down via the hi press switch. Self A/C charging with cans usually this culprit. Gauges will sure help, hi and low numbers. Any shop would charge you a minimum just to check the numbers.Well worth it. NOT a green and red hand held from Pep Boys etc.
Could your expansion valve be getting dirty, and not flowing. Just thoughts.