W126: engine shocks question

Have an '88 420 SEL with 170K.

Recently, I had engine mounts replaced on both sides - as one appeared crushed.

However, I did not replace the engine shocks at the time.

I still feel some pronounced left to right movement of car upon starting up, but only for the 1st 1-2 seconds of initial start-up/cranking.

What exactly do the engine shocks do? And should I have them replaced?

Thanks.