I read through here in great interest because I built a lot of these differentials in the past and just had one last summer here where I'm retired.
That the vehicles get a 'shimmy-turn' or a 'click-pop' noise and the rear end moves sideways was always a sign that the oil in use in the differential was OK, but missing an important ingredient.
I had a '73 El Camino, 350 SB, factory 4 sp, no A/C, just PS and PB.
It had a 4:10 Posi rear, factory dual no-cat exhaust 4" system all the way back with no cats as they were an option in those days even in California.
I put on Dual Q-Jets remounted on a 1969 Camaro Z28 X-over manifold.
With 13:1 compression ratio, Mallory Dual Point Distributor, Chet Herbert solid roller cam, etc I had to run 120 Octane (the old octane method number, not this new fake numbering system).
But the biggest problem it ever gave me - besides a couple of pursuits that I won (and the statute of limitations is 'way gone by now!) was that the GM Posi-Traction additive died at regular intervals and needed refreshment to keep the diff from alternately grabbing and releasing.
After making 20 right turns and 20 left turns in a parking lot one day so I could travel on for a while until the clutches either squeezed the additive out of the plates or whatever - well, it was embarrassing! I did that in The Tyler Mall off the 91 FWY in Riverside CA., and I scared small children and old ladies and hurriedly got out of the lot before the security cops got there.
But by getting it back to the shop in Santa Ana and replacing the lube with Castrol 85/120 EPW and a bottle of GM Posi additive, it was good for a year or so.
Depends on how hard I pressed it - but after one night at the drags, I'd have to change the fluid before I could drive it home again.