Fans not working right

When you turned on the heater, that just naturally opened the water valve on the heater core that is full of cooler water than what's already been warmed to operating temperature. Automatically the sensor is going to note this change and turn the fans on and off until the coolant is at operating temperature again. Nothing to be concerned about, it's doing the only thing it knows to do. That's what it's made for.

Rather then rigging it , fix it right ! The blower motor could be bad . Testing the electrical circuit , checking current draw , voltage drop . etc.... Hooking up a light switch isn,t going to keep it on if parts are failing . More then likely the blower motor control processor is going bad . The blower motor is electronically controlled .
HVAC Blower Motor Circuit

https://www.1aauto.com/2006--pontiac-grand-prix-blower-motor-module-for-models-with-automatic-temperature-control/i/1ahbr00026?f=489143&y=2006&utm_source=bing&utm_medium=cpc&utm_campaign=Shopping%20Campaign%20F-L%20-%20Medium&utm_term=1100309270240&utm_content=F-L

Check the wiring to the motor and resistor pack For the motor to run at full speed all the time indicates that there is uncontrolled voltage to it which means that there is a earth after the motor before the resistor. For the resistor to burn out .Well it requires fan air flow to keep it cool and with the ignition off it must still be getting current to it to earth. I am suspecting a faulty automatic control module. I suggest you have it check out for failure.

you could apply 12v battery power directly to the fan itself...there should be 2 wires coming from the center of the fan, follow these and you should come to a plug that you can un hook as if to remove the fan. on the fan side of the wires, hook them to 12v power, one wire + one to - if the fan does not turn, switch the wires polarity and test again. ( the fan should run either way, polarity just dictates direction of rotation) if the fan spins, the problem is on the car side. if not the fan itself is suspect. test vehicle side with known good fan or test light.

Try by passing the switch and see if the fan runs. Also check for power at the switch. If you have power and the fan doesn't run it's the fan motor. If you don't have power change the fuse and try again. If it does run keep an eye on the fan it could be going bad.

Have you replaced the right switch that mean the switch with the right temperature according for this engine.

it's controled by a fan switch, when it get to a certain temperature the switch closed and power the relay which makes contact to feed power to the fan.
you can do a toggle switch like that, make sure you have right size of wires and the switch can handle the load, and the fuse is the right amp.
the possible damage is that if you ever forget to turn off it will drain your battery and make the fan motor go bad quicker.

have you checked the fan switch which should be on the radiator, maybe faulty and operating at the wrong tempreture.

door switch

The multi function switch is the turn signal switch. It also does control wipers and lighting. If you are having the issues you describe then you will most likely have a body control module problem. Before you spend the money to buy the module and get it programed get it checked first. Goot Luck