My 2011 HHR failed inspection because when the brakes are applied the drivers side tire moves back and forth, front to back movement. When I purchased it there was an issue that I didn't think was as bad as the tire movement appears to be. When you accelerate, or hit the brakes the care has a slight sideways jaunt or slide if you will. Obviously these two are related. Im just needing a little direction so I can troubleshoot not change parts.
You need to check the suspension joints as a failed bush can allow for the movement you describe.
Inspect the tire tread as a failed bush can cause rapid tire wear on the inside or outside edges.
I have just replaced all of the rear suspension on my car as one bush had failed after 15 years - logic being the others were likely to fail soon. Result is a car that has no squeaks, rattles or clonks and handles like new.