This is the fan motor & the resistor is magically replaced inside the motor as in pic 2. It is now called a speed controller
I don't think the motor is the problem in this instance. There is another module at the passenger kick panel area.
Did you inspect the relay plug for overheating? Sometimes they can melt the plug or the load is shorting to power.
You should be happy the motor operates, they are hi-$$$ and commonly fail.
I just don't know if the 2 power wires are normally hot at all times. If so, the motor may be the problem.
The smaller wire in the center is what controls the motor function.
Others may be able to shed more light on HVAC troubles.
Actually most likely is the speed controller in the second picture^^. The power wires are hot all the time(orange+ and black -). The thin wire in the middle is the one controlling the speed based on the voltage value (between 4V and 9V). Is possible to have a component shorting out and blower stays on even with no control voltage.
Unplug fan connector and measure the voltage between the black wire and the thin wire when the fan should be OFF (harness side). If no voltage then the Heater and A/C programmer is fine, the blower motor is not. If you have voltage (when you shouldn't) then the motor is fine and the Heater and A/C programmer is the culprit.
By Heater and A/C programmer I understand the module underneath the glove compartment, NOT the module underneath the stereo.
You need the electrical diagram to diagnose this, is not a conventional resistor based blower motor control.
But no matter how scientific you get, in the end 90% of the time is the blower (either motor or incorporated control electronics).