[Coupeguy2001]

I noticed that the spline in your pic is a bit rusty.
I would take it apart, clean it, inspect for wear, and grease it going back together.
Maybe the vibration has something to do with the spline joint not sliding in and out or the rubber in the center bearing support is old and weak, and allowing the driveline to move around?
Sometimes old shock absorbers are sticky in their movements, and bind, causing the rear not to float with the springs as designed.

I just changed the shocks on my tahoe, and sure enough, one of them wouldn't compress. I even put it under the jack point on the rear of the car, and the tire would not touch the ground because the shock was holding solid. Even with the weight of the vehicle.

Would seem to me that if it was designed to be there, then the change that takes place is wear, contamination and age.

1. wear
Check the U joints one at at a time by pulling off the caps, and checkiong the stub ends for wear.
Check the splines for wear
check the trans spline for excessive wear and in the yoke.
2. Contamination
Check that the little boots on the u joints are really sealing out the water, and keeping dirt and other things out.
check the grease in the splines by feeling the lube as it comes off the splines and u joints.
Check for any water getting in where it is not supposed to be.
3. AGE
As we all know, parts that are moving, wear. the wear is exaggerated by the things I mentioned above.
Sometimes parts like rubber and leather, grease, dry out and do not perform like when new.

Like on the cooling system..... Just keep trying things.