The rear shocks seem to be very firm and I think it has an air pump to inflate

could be the shocks.. Remove the shocks- fully extend, them plug the opening that the the air lines hook to .. Fill a tub (that will fit the shocks ) with water- submerge the shocks in the water and compress them - if you see bubbles them you know that they need replaced.

it's the rear shocks ! , i have a 94 park avenue and it had the same problem the air pump ran more and more and the rid was awful ! changed the shocks and the car rode great ! and the air pump didn't come on all the time