How do fighter jets stop pitching motion after a elevator deflection?
This depends on what level of realism you want.
If you want high, save yourself a lot of work by using an existing dynamics model like the JSBSim. There are model templates for it that have all the needed functions for reasonably realistic model are predefined and you just tweak the coefficients to get the performance you want.
If you want low, just make the surface deflection correspond to pitch/roll rate rather than acceleration, because that's what actually happens to a level of approximation that is perfectly fine for an arcade gamer.
What really happens is that when you deflect the control surface, the rotation will change the angle at which air hits the surface, and that will negate the effect of the control surface, so the aircraft settles at fixed rate of pitch/roll corresponding to the commanded displacement. And when the pilot centres the surface again, the aerodynamic force still exists that countered the deflection, which will now stop the rotation again.
This is fast enough that pilots normally don't need to stop the rotation with counter-movement of the controls except sometimes for fast rolls in aerobatics.
Regarding PID controller, a “position-integral-derivative” controller, the point is that the derivative term predicts the response delay, while the integral term compensates for any bias. The disadvantage is that if you tune it to have fast response, it always overshoots the target value, because the integral term will overestimate the bias. But it is simple to implement and does not require much modelling, just tweak the numbers until it works. The JSBSim provides PID controller (and some other simpler controllers) for the model definition.