Mustangs are not known for their long suspension travel, so bottoming out is fairly common if you have a heavy load in the car. It is unusual to do so if its just the driver in the car.
Replacing '79 and up rear springs is easy on a live axle car.
Raise the rear of the car and support it on the chassis with jackstands. Place a trolly jack under the diff , and lift the axle about an inch from full droop.
Remove the rear shocks at the lower mount only, and the rear wheels. You may have to loosen the rear sway bar, if equipped.
Place a jack under the lower control arm and remove the pivot bolt and nut, one side only. Lower the jacks a bit (be careful not to stretch the brake flex line), and the spring should come out.
As for the side to side movement......inspect the upper control arm bushings for wear and looseness. Since you are already doing this work, consider removing all the control arms and replacing all the bushings.