It is great that everyone is sharing their particular experiences. The items that have failed for them are certainly items that also may be causing your problem, however, it is impossible to know exactly what is causing your particular problem by just reading your description.

Badly worn shocks could be the cause, loose rear suspension links, very low tire pressure(s), sway bar bushings, suspension joints of all kinds, steering play, etc., etc., etc., ............

You need to raise the car safely and start inspecting all suspension joints, shocks and everything. My approach is using the 4 P's, Push, Pull, Pry and Prod on all steering linkage and suspension components.

Good luck,