Cobalt SS suspension problem

I was just going to say the same thing. Check your motor mounts.

If you have cable operated throttle body, open your hood with the car in neutral and rev it, see how much play there is in the engine.

If that doesn’t work, have some one get into the car, bet on the brakes, put it in gear and slowly let out the clutch to put the load onto the car while still on the brakes, to see what’s going on.