surely the shocks are cooked to act like that

go back to basics, a stock car would not corner like that, something is worn. Bushes and shocks is not expensive. You should not require additional components (swaybar/panhard) to correct this issue

disconnect the swaybar to see if it's any different, but if you at all suspect on the shocks then replace

It's not something that's happening over time/temperature, it's happening from go. So if it's the shocks they are totally rooted