Noise in rear of 2003 Buick Century

On a 2006, the towing weight probably split the Sway Bar bushings. The sway bar stops the back from swaying. Could also be shocks as well or both.

Push down on the back and see if it bounces.

sounds like you have a brake pad hanging up your shoes probably need to have brake shoes checked and adjusted

Are any of the speakers loose?

If you have fitted new shockys I suggest you check the travel of the units as it sounds as if they are bottoming out Pistons hitting bottom of internal cylinder.

check the u joints in the drive shaft or the rear end needs greased,or it could be going out.have you changed both the inner bearings as well on the front end?

most likely a blown shock/strut or mount bushing

Im not saying this is a definite answer, but since I worked at a dealer 90% of these cars had an issue with the lower control arm. it is exspensive but by simply haveing a person watch your tire movment as you shift from fwd. to rev. with your foot on the brake you will see the tire actually move back and forth when you shift it.

Get the codes pulled. The check engine soon light means there is a issue that needs to be addressed. Once you have the codes pulled you can use them to diagnose and locate the issue.

You might have broken the sway bar link, or the strut, or strut mount.

you have a bad front wheel bearing..jack up front of vehicle..put in neutral..block rear wheels..put 1 hand on front coil spring and rotate wheel..you will feel roughness in spring on side that wheel bearing is bad