What will worn shocks do to my car’s ride?

If the rear shocks on my car are bad but not blown, can that cause my car to sag or sway on turns?

The function of shocks is to control a vehicle’s bounce and sway. Bouncing and swaying become more apparent as shock absorbers deteriorate. If a vehicle has very worn out shocks — or effectively no shocks at all — it will bounce along rather than ride smoothly.
Yes, worn shocks will not be able to absorb the sway and bounce induced by changes in road conditions such as bumps, holes, curves, and turning.
The springs will bounce as the movement’s energy is absorbed by them instead of the shock absorbers. Shock absorbers are intended to absorb the movement’s energy.