Can I just cut the wires between the circuitboard and potentiometers, and solder the wires to the hallsensor instead..?
Yes, but observe that pinout in HALL sensor is different from pots.
Pots: +5v - SIGNAL - GND (invert +5v wires and GND to inverted response).
HALL (A1302 - front view): +5v - GND - SIGNAL (invert wires to get inverted response is NOT allowed).http://www.leobodnar.com/products/BU0836X/A1302-BU0836.gif
So, check the datasheet of HALL that you buy for correctly pinout.