Adding 500ohm after Low pass filter helped, there is no more noise from 0 to about 3.73v.
It seems to me that after that 3.73v its near its saturation range, and that noise seen in second photo can be causes by power supply trying to limit current?
The mosfet is going to change rapidly near the threshold.
See the attached plot - notice the drastic changes from 2.4/2.6/2.8V, and that is before temperature changes kick in.
And yes now it looks like i need a feedback from shunt and opamp so it can do its control loop.
Btw, ive built this simple circuit just to test the theory about mosfet acting as "adjustable resistor".
Feedback on a multi loop system, is very hard to get stable, and you have a very large filter capacitance there.
A MOSFET will work more as a current sink than a resistor, and will need mV steps on the gate.
It will also change quickly as it heats up, so you need very good cooling.
Those are not unsurmountable, and you may be better to work on a design that tracks thermal changes and sets a target load current at a calibrate step.
Such a design does not then care nearly as much about the load behaviour.
You can use NPN transistors or MOSFETs as the load.
A benefit of NPN transistors, is an accurately defined base current, gets you a good starting point.
See the 2SC5200 plots below, that's a 'good' NPN part, with high and flat HFE.
If the MCU measures temperature and sets precise base current, you can knob-set a test current, capture the Base current, and then apply a small temp correction, as it warms up.
No multiloop contentions to worry about.