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Your shocks, which should more appropriately be called dampers (That's what the rest of the world calls them) dampen the oscillations from your cars springs. Basically, if they are not working right your car will bounce around on you.
If you have not had your car aligned in a while I would definitely take it and and have it properly aligned at a good corvette shop. From the way you describe things sound like you are getting torque steer.
Comparing hard old tires to new tires, everything I run over now is quiet... I literally felt every little pebble with hard, old supercars. New PS2s absorb all but the harshest stuff. Even when cold.
Designer Imagines A Corvette That Looks More Like a Corvette Than the Corvette
Slideshow: A Jaguar designer's personal project imagines what a modern front-engined Corvette might look like if Chevrolet revisited the golden age of the Stingray.