Why does my low speed fan on my 2007 mustang intermittently work

What is the pressure in your low pressure side at 80 degrees ambient? Did you use gauges?
U said the clutch will not stay engaged, meaning that it engages and then releases. If so, that means the clutch winding is good but that perhaps you have a loose wire feeding the clutch or a bad, intermittent relay. And if that is not the case then you are LOW in freon and the low pressure cut off switch is disconnecting the clutch when the pressure goes down. Pressure should be around 35 pounds on suction line at about 3000 rpm.

You may need a resister for the heater fan motor.

On any other vehicle it sounds like the blower fan switch.
The fan switch provides power or ground to the resistor and there would be a high speed blower relay to bypass the resistor.
But in your case the fan switch is part of the controller.
I don't have a diagram for a 92 but the 94 model shows relays for low medium and high speeds. Each relay gets power from a fuseable link. The high relay gets power thru the low relay if the low relay is not engaged.
If you have the model with 3 relays you may need to check them.

Hold the idle at 2000 RPM for 2 minutes then drop it to 1500 for 2 minutes and then 1000 for 2 minutes, then let it return to idle for 5 minutes. Full relearn requires some driving as well, mixed low and highway speed and idle time.

There are three wires to your electric fan; a ground, and low/high speed power wires. You can try jumpering the low speed power wire directly to the battery to see if the fan is bad or not. If the fan works it may be a bad relay.

I would check the heater blower motor resistor.

check the reostat which is a resistor that controls the fan speed and also check the wiring to both the reostat and the heater motor

is the temp gauge working? sounds like coolant temp sending unit... or a possible engine coolant temp sensor or ect sensor.

do you mean the fan speed?there is a resistor block that needs replacing that controls the speed or the fan is no good. connect the blower fan to 12 volts after unplugging it,and see if it works properly

ok. so you know the fan is good. switch out the relay for that fan with another same relay and check with the a/c button.