Transistor buck converter system current increase
IRF520 needs 10V on the gate to fully turn on. Your Pi outputs 3.3V, which is insufficient. The FET goes in linear mode. As it heats up, its threshold voltage gets lower, which explains the increase in current.

Solution: use a FET which can be driven from 3V3, or use a level translator to make a 12V drive voltage from your 3V3 control signal.

Also, it is unclear why you use two FETs. Since they switch low-side (in the GND wire) and you connected both FET board's grounds together, at least one of them is useless.

EDIT:

Your new module has a MOSFET driver which accepts logic input, so it should work. I don't know what the "VCC" pin on the input side should be connected to, though. 

Also you still can't use 2 low side FETs in series, as the GND pin on both boards need to be tied together for the control signal to work, so the first MOSFET can't interrupt the GND of the second one...

Much simpler to ditch the MOSFETs and use a 700mA LED driver with an ENABLE input. Just pulse the ENABLE input high or low to control the LED.