Doesn't sound to me like front shocks. If your front shocks were shot (which might be the case as well), then you'd have a soft, boaty feel, it wouldn't handle well any unevenness of the surface, it might get bouncy as well (but if you don't know what new shocks feel like, you might have no point of comparison, new shocks make a huge difference in the way the car handles). Your symptoms sound more like upper balljoint/ control arm issues or possibly lower ball joint. Maybe just some bushings, if you're lucky. Does it make squeaky sounds when you turn the steering wheel extremely to one side? Lower ball joint is not really a DIY job. Upper control arm might be ok to do yourself, but not as easy as shocks.

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1974 240D, 125K mi, B100 now, SOLD
1980 300TD, 115K mi, 4-speed stick, roll-up windows, greasecar with two tank conversion (daily driver)
1985 300D, 220K miles, greasecar with two tank conversion, SOLD
1993 300D 2.5turbo, 158K miles, green/tan, B20-B50