This a solid rear axle vehicle, and if so there is a panhard bar that secures the rear diff (axle) from shifing left and right mounted to the frame, oops I mean body? Check every piece of it if it has a center pivot link for anything broken, loose or worn. Worn or broken rubber bushing cause a lot of suspension issues your mentioning and it doesn't take much especially with soft rubber to cause mushy body control.
While uner the vehicle have someone, a larger stronger person than your 92 lb. housewife. Push & pull the body left and right rather hard while your looking for loose rubber joints in the system overall.
Someone with abilities just above able to change a tyre should be able to locate your issue along with a King Kong helper body shaker.
Independant rears require more links to secure each wheel from shifting left & right and front to rear unless your "swaying left and right" which is body roll each direction when turning left and right driving a "S" twisty road at speed?
Even replacing the soft OEM ($$$) rubber suspension bushings that are soft for a smooth ride they will go soft and fail again over time. With Poly bushings (if available?) you'll have sharper handling, less anti-roll plus increased steering sharpness with steering wheel feedback. Slightly firmer reaction results that outlast rubber.
To me when I read "sway" that is an issue with the "anti-roll" bars be it at their mountings as well the link connections again talking rubber that has gone soft or falling apart an issue that Poly replacements prevent over rubber the only replacements I install on my vehicles. Poly vs OEM rubber will outlast the vehicle.