[Sirrotica]

Seeing the picture, it's a lot of play, but unless someone has been in the axle, and set it up wrong, there is likely something worn out, bearings or gears would be my guess. Plenty of reasons to get excessive backlash other than not shimming the ring gear deep enough. This may, or may not, be the noise you're experiencing at low speeds.

No matter, excessive backlash isn't going to cause any problems that translate into poor drivability. If you don't have the necessary tools to set up a rear axle, I would suggest farming the operation out to someone that does, and is totally familiar with setting up rear axles.

Pontiac cars have always been geared really low numerically to get decent fuel mileage (2.41, 2.56, 2.73) ratios are fairly common, so unless you have a fairly radical cam in your engine, the rear axle ratio isn't something that's a consideration in a drivability problem at low engine speeds. Most engines can pull smoothly under 1500 RPM without bucking, and jerking.

Thing is no one here that can lay hands on your car, so trying to do internet diagnostics is just shooting in the dark, basically guessing. In the past people have posted videos of noises here, but just hearing the noise on a video isn't going to help much without actually hearing it in person, then trying to find where it actually is originating (stethoscope, long rod, hose held to the ear). Much easier to do diagnostics where the car is.

Hopefully you can find someone locally that can give you a hand sorting all this out.