My 86 300E did exactly what you described. In addition to the suggestion made by Steve, you should also want to check the subframe mounts. I believe there are posts elsewhere in this forum dealing with subframe mounts.
My mechanic replaced the thrust arms (the links that Steve mentioned in his reply) and the subframe mounts and I replaced the shock absorbers and the problem is more or less fixed. I believe I need to have the rear end aligned to finish that part of the fix altogether.
Folks more educated about this than I may be better able to respond to this, but I believe that the job is also not completed unless the front struts and front-end alignment have been inspected. A loose front end, it would seem to me, would contribute to the loose feeling all around. Maybe someone could also comment on any contribution that the springs might have to the "fish-tailing" effect. I'm considering replacing the springs as well.
1986 300E - "Liesl"