Turned out to be the rear vertical mounted control arm bushing was bad. Mine was on the right side. They are difficult to inspect, you need to get the car up on a lift and put a bright light on the bushing to see it. Don't let the shop tell you there are no after market parts!!!!!!!
The easiest solution is a new control arm Dorman 521-025 (L) & 521-026 (R) for FE1 and there are a couple of other mfgs, they include new bushings and ball joint RockAuto and EBay around $68 + shipping. R & R is a matter of 4 bolts. You will need a torque wrench for install. 1 hour per side including jacking it up.
You do need to be careful when ordering, the parts listed for "FE1" are for Cobalts that have a powder coated steel control arm V. HHR control arms are cast and don't seem to be listed by package, except I did see 1 mfg advertising "new" FE3 for alot more $.