With a non heat pump it is a simple 24 volt AC control circuit. In the attic there is a relay that is actuated via the fan control wire directly from the thermostat. On the thermostat there is an auto/man switch that controls the fan. Manual the fan runs all the time. Auto fan turns on and off with the outside compressor.

If the system is not calling for cool and the fan continues to run the fan relay is sticking. common problem. Sometimes after running for several days the relay will stop sticking if not, replace relay.

If the system is calling for cool and the compressor stops I would still expect the fan to run until thermostat tells the A/C to off. Some compressor units include a safety switch or high-pressure cut-out switch A blocked internal valve bad control board or external fan in outside unit defective no air flow in outside unit.
High pressure cut out not good,

Suggest turning temp max cool and observe. If acceptable cool and the compressor runs without interruption all is good, except for the blower in the attic. Safely check relay in attic blower circuit.

Hope this helps.

When you say fan still run is it the inside unit fan .if it is then issue is with the unit outside