The best way to tackle this would be using rotational shake in your camera script like this:
set_rotation(Vector3(deg2rad(pitch),0,0)"
set_rotation(Vector3(0,deg2rad(yaw), 0))
set_rotation(Vector3(0,0,deg2rad(roll))"
where yaw/pitch/roll can be either random values (shaky af), pre-generated noise or plain sin(time) for a gentle sway.