Hi, Gia!
I take it this A/C has the two knobs on its inside housing. One's On/Off/Fan/Speed and the other is Temperature? Not ducted, and no wall thermostat? Fan runs full time and Compressor cycles for Temperature?
If so, that Temperature Knob is directly mounted to a bulb type thermostat. I don't think it can be much wrong but something with the Thermostat itself.
Knob goes on that shaft, and the coiled part is the temp sensor. Mechanical contacts inside cycle compressor according to temp experienced by the coiled piece.
Sound like what you have?
Please post Make/Model of the A/C.