Do I need a new rear end?

Drain the gear oil and see if there’s any chunks of metal. And stick in a spring magnet towards inside bottom to see if any metal comes out also.

 

Power takes the path of least resistance. If the brake on the wheel that is hardly moving is dragging a bit, its easier to spin the wheel on the other side (which will spin twice as fast). Normal differential action and not neccessarily cause for any concern.

I didn't watch with volume so I cannot say anything about the howling.

 

Do the easy stuff first, drain both, examine what comes out. Refill both with synthetic 75w-90. See if problem improves, worsens, or no change.