An idle speed of 600 rpm would be normal for a diesel, but for a gasoline engine it is a bit low.
Because of the low speed, the engine may no longer run smoothly and the whole car shakes, which is strongly noticeable on the gear stick.
There is probably a defect in the speed stabilization / idle control.
A defect in the dual mass flywheel would also be possible, which also has the effect that it is difficult to start off without jerking.