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).

So, check the datasheet of HALL that you buy for correctly pinout.
A1302: http://www.leobodnar.com/products/BU0836X/A1302.pdf