I actually own one of these beasts as well.
Front end problems are very common with this solid-axle holdout.
The steering dampener does absorb some 'bump-steer', but otherwise just masks any underlying problems with play in the steering. Larry made some good suggestions (as always).
Here are some more.
1) Make sure the steering box itself is tight. Look for excess play before the steering arm even reacts. Adjusts just like a 124, but in the reverse direction! Made a big difference in my case.
Check the steering box bolts and bracket as well. These are known for coming loose, or even cracking. Mine has been ok.
2) Check the Trackbar joint and bushings. Joint (frame end)has a grease fitting, so it will last *forever* if lubed and the boot is intact. Bushing end (axle) is a common wear item.
3) Steering linkages are an obvious potential source of trouble. They all have grease fittings, so they will last if greased.
4) Control Arms (4 of them) have bushings that can wear out. They last pretty long, but at 147k...
5) Wheel hub assembly pivots on what is essentially a fixed hinge(solid axle, so moves in just one plane) with a top pin, and a lower pin. The top pin has a grease fitting. These things are not generally the 'weak link', but that is a lot of years and miles.
6) Swaybar bushings wear out even faster than on an MB! Not really a "shimmy' issue, but does lead to general instability.
7) After carefully going through all of that other stuff, my biggest Death Wobble contributor was found to be a worn hub bearing. It had significant play, but made zero noise. These are pressed one-piece hubs (like on a typical fwd), which cannot be adjusted to set preload. These things are known to have a short lifespan. They are expensive, but easy to replace.
The whole front suspension is very primitive, but also relatively easy to service. No worries about camber settings! When all front-end components are in good shape, it handles better than one would imagine.
1986 300E 5-Speed 240k mi.