Here's a couple different things for you to check out.

The sway bar pivots in a pair of rubber bushings. They get old, worn, & the hole gets too big. Then the sway bar can rattle around & not control the suspension properly.

Maybe one of those pivots has even come loose from the bottom of the car?

One of the end links (connect the sway bar's ends to the lower control arm) is loose, broken, disconnected, or ???

It's possible that what you feel may actually be your shocks getting soft.

Or maybe 10 years ago a bunch of paint got knocked off the sway bar & it finally lost the battle with rust. Seems not very likely considering how thick the bar is.