I have eliminated rims as an issue. When I pull and tug on suspension leakage all seems to be solid except for the radial arm. When I pull forward and back on the arm I get some play and the backing plate moves a little bit. Would this be the problem. Also wondering since the tires are real old would that be the issue. Lastly is there a progression one follows to identify what might be the issue. Hence to or replace first?? I am new to this so any help would be greatly appeciated. Thanks, Bill

The worn radius arms would probably not make it shimmy as the weight of the car kind of takes care of that. Would have an effect on the caster. The first thing I would do is install new tires.

I would go with new tires, too. I had a similar problem and new tires took care of it.

i would go with new tires too.
if they sat for a long time they might have flat spots on them that you can`t see with a naked eye