PSUs click and fail to supply power
Since the question has changed, here is a new answer.

I won't state the obvious, that each 3 Ohm resistor would need to be 48 Watts minimum, preferably 100 Watts, which means it needs to be a huge wirewound thing. ... Or a 50 Watt 12 Volt lightbulb. If the resistor is of lower wattage, the PSU won't fail, the resistor will vaporize when powered.

Also, the question does not specify which MOSFET is being used. A datasheet would help figure out whether the MOSFET in question is capable of handling the load at all, or has melted inside and bonded into a short somehow. 

Sequence of steps to test:


Preparation: Check if the ground line of the PSU and of the microcontroller are connected together.  If not, make that connection. Also add a hefty (1000 uF 18 Volts minimum, more capacitance is better) capacitor between the supply and ground, as close as possible to each set of resistor and MOSFET, + side at resistor's connection to +V, - side at MOSFET source connection to GND.
Power up one PSU with just one 3 Ohm resistor connected between +12V and ground return lines on the PSU. Does this cause the same click/overload? Then the PSU's current capacity is less than 4 Amperes, or current limiting is set for less than 4 Amperes.


Is the resistive load getting warm / hot? If not, it is most likely not passing the 4 Amperes calculated from the resistance. 

Power down, add a second 3 Ohm resistor in parallel (+12V -- Resistor --  GND), and power up again. Click / overload? Then the PSU cannot handle 8 Amperes current, or current limiting is set for less than 8 Amperes.


Are both resistors getting hot? If not, the PSU voltage is sagging due to overload. You need a more powerful PSU.

Power down, hook up one MOSFET with gate and Source both hooked to Ground, Drain to  the 3 Ohm resistive load, as shown in schematic at the end of this answer, power up again.


The resistor should not get warm this time, the PSU should not have flickering status lights. If the resistor is warm, the MOSFET is shorted, or is hooked up wrong. Check the datasheets, check the pin-outs.

Power down, add the second MOSFET (gate shorted to ground) and resistor, power up again. Should have same results as above.
Power down, disconnect the MOSFET's gate from ground and connect it to the microcontroller's output through a 100 Ohm resistor.
Fire it all up, output a 50% duty cycle PWM from the microcontroller. The resistor should get warm, but not as warm as step 2 above. Hook up a multimeter in DC voltage mode between MCU output and ground. It should read around half the microcontroller's supply voltage. 
If any of this fails, then share the MOSFET datasheet, the PSU datasheet, and a photo of the hooked up circuit.

   



   


simulate this circuit &ndash; Schematic created using CircuitLab