Right, NOT worn out.
These old engines complain a lot before giving up the ghost. Hard starting, low power, oil consumption are the signs of the end. Your engine, despite its 300K miles sounds like it's in good shape - no doubt due to good prior maintenance.
I'd suggest you dump the mechanic who claims the engine is tired. He is looking at its mileage rather than at its condition. In his favor is that most engines of this mileage would need an overhaul by now. Yours is the exception that may go to 400K if you take good care of it.
By the way, with a 10 mm and either 12 mm or 14 mm wrench you can adjust the rack damper pin. It looks like a small bolt screwed into a nut (lock nut) on the aft end of the engine's injection pump. Look between the oil cooler lines at the back end of the injection pump.
The larger nut is a lock nut, the smaller bolt is the adjustment. Loosen the lock nut one half turn. Then screw in the bolt on half turn, hold that position with the wrench while you snug the lock nut tight. By tight I mean firmly snug, not tight as in all your might tight. This is an incremental adjustment, more may be needed but first drive the car for a while. Too much tightening will stall the engine, if so, back it off. If the bolt is screwed all the way in and no further adjustment is possible the pin ought to be replaced, the newer version has a stiffer spring and is more sensitive to adjustment.
My 300SD used to shake - a lot, particularly at a stop light after exiting the freeway. This adjustment and a new pin completely cured that.
Finally, the engine's shaking won't harm the transmission or its mount.