Controlling MOSFET's resistance using negative feedback
Your feedback system is two op-amps in cascade followed by a MOSFET. The first op-amp is unity gain and the 2nd op-amp is used as a comparator. The MOSFET will have high voltage gain because the source is at 0 volts and Vgain = Rd/Rs so it's high!. Additionally the input capacitance of the MOSFET (about 1 nF) will likely turn the driving op-amp unstable too.

This will inevitably cause stability issues because it's quite easy to make an op-amp go unstable by making a complex feedback loop involving an extra comparator and a high gain transistor.

You need to lower the gain of the loop and this might be acheived by making the comparator into a non-inverting integrator. You might also try putting a resistor in the source of the MOSFET - as high a value as you can tolerate.

Unfortunately the LM358 (old school device) doesn't have a proper graph of its phase margin so it's tricky to calculate values but for the integrator 10 kohm and 1 nF mightd work so I'd be tempted to ditch the unity gain op-amp and feed "ADJ" straight into an inverting integrator.