Sounds to me like the motor mounts are loose in front and the sub-frame loose in back, and the only thing keeping the drive-train in tact is the center drive line support and possibly the transmission support. Hence, all your vibration is being transmitted through the good tight points of contact. You are always going to have some slight unbalance, but it all magnifies with bad mounts.

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1976 240D
1987 560SL
2007 E320 Bluetec
1998 C280 (now son’s car)
1982 240D Manual - Sold