Yes, you can do both, switch between two different sense resistors and between two reference voltages.
In the first case use two very low ohmic MOSFETs to connect one of both sense resistors at a time to signal ground, which would mean to connect the drain of switching MOSFET to terminal of sense resistor which was formerly connected to signal ground and to connect the source of this MOSFET to signal ground.
You can also have one sense resistor permanently connected to signal ground and only one sense resistor switched to ground, which then is in parallel to the other one.

Disadvantage of this methode is, that you need a MOSFET for this switching task, which shows a turn-on resistance that is much less than the sense resistors.

For this reason I would switch between two different reference voltages: Use two different trimmers, one for each reference voltage, and connect one of both at a time to the L6506 via CD4066 switches. Directly at the Vref input of L6506 connect a cap to signal ground to prevent switching noise from making trouble. A little RC low pass filter here would increase the benefit.
If the turn-on resistance of CD4066 switch is too high (should be much lower than the impedance of trimmer), then you can also choose the 74HC4066, or an analog switch like DG419 or else.

Kai