You have a feedback loop stabilization problems here. The common source MOSFET adds voltage gain inside the feedback loop and presents a capacitive load to the operational amplifier. Operating with gain helps but at the very least, an output capacitor is needed but this is one of those cases where oscillation is likely if the ESR is too low. Similar integrated voltage regulators designs often place a resistor in series with the output capacitor to control this. Pole-zero compensation with a series capacitor and resistor from the operational amplifier output to the inverting input will help but you will have to modify your reference filtering network. The circuit would be easier to compensate if the gain was fixed and the reference voltage was adjusted.
Compensation can be checked by doing a transient response test.
Since this is a low dropout design without a bias supply for the MOSFET gate, I assume you drew the arrow for the p-channel MOSFET backwards.
From a noise and drift perspective, the OPA227 is the right amplifier for the impedances shown but its high bandwidth exasperates the feedback stability problem. An OPA177 which is slower might be better and would allow easier reference filtering. I like using LT1012, LT1097, and OP97 style operational amplifiers because they allow easy adjustable overcompensation but I am not sure if TI has an equivalent. Their even lower input current would allow for easier reference filtering.
If precision is important, then leakage in the reference filtering capacitors may be an issue but the film or ceramic capacitors you show should not be a problem. Their their small size means they are not doing anything for low frequency noise though.
I would probably put the variable resistor in the low end of the voltage divider so if it fails open, the output voltage drops to the minimum. Grounding one side of a potentiometer often has noise advantages although that probably does not apply here.
If the output current is not high, I would use an NPN output transistor with a unity gain level shifting network.
As shown your circuit can only source current. If you want +/- 100 milliamps, then you need a different design.