In my wagon, my idle rattled like hell, it got better when you were driving but not by much. Shutting off the engine rocked the car some my trans mount was shot too so, all that was wobbling around etc. It could be motor mounts. It's certainly an excellent place to start. I'd do the shocks and mounts and see what rattles next. There could be a couple bushings in the front steering that could need attention too.

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hum.....
1987 300TD 311,000M Stolen. Presumed destroyed