Air Conditioner Turning Off When Thermostat Lowered

The air conditioner on my 2003 Mustang V6 functions fine until you turn the thermostat up to not get so cold. the compressor turns off and on to keep the temp comfortable. Once you need cooler air and turn the temp knob counterclockwise to lower the temperature, the compressor turns off and will not turn back on and cool. What is my problem?

I have noticed when I stop the car, turn the A/C off, kill the engine for a couple minutes and restart engine and A/C that the air compressor restarts and cools.

The compressor clutch will not engage at all today so I tried to charge the with more R134a. The system would not take any more freon. Thanks for your failed recycling switch suggestion. This afternoon I first checked the low pressure switch up near the firewall by jumping the electrical on connector, cranking car then turning on A/C to Max Air. The compressor clutch would not start the compressor. I understand the high pressure switch should stay made or closed to allow current to flow when you have normal static freon pressure. I assume I do since the system would not take more freon. I checked continuity on the high pressure switch down low on engine below compressor on high pressure line and this switch was closed making connection saying the compressor clutch should be turned on. Is my logic good for my 2003 3.8L V6 Mustang?

After a weekend out of town I next planned to check compressor clutch as next step; however, A/C is working fine today. After interior of car cooled on Max Air I repeated the process when A/C failed. I raised temp by turning knob clockwise 1/4 turn, drive a few miles, then turned back to Max Air. The A/C cooled great. Next I tried various up and down step in temp and A/C did not quit performing. I am sure problem will resurface. Any more suggestions? I have considered too much moisture in the refrigerant as the possible problem. I plan to test it on a longer trip before weather gets too cool this fall.