It sounds like excessive backlash in the ring and pinion to me instead the Detroit locker. The Detroit will make quite a sound if there is excessive backlash when it locks in.
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I am guessing the slop has to be between the axle splines and the Detroit carrier then.New R&P, when i felt the driveshaft slop of 1/3 of a turn i thought the same thing. Damn! I blew the backlash adj. Pulled out the indicator and checked. .008
? ? ? The 1/3 driveshaft turn was before it started to turn the R&P
Then i saw that it's common on the Eaton faq page

Yes, there has to be some pre-determened clearance between the square cut teeth so that the locker can lock and un-lock to differentiate, or it would never unlock.For the record, you can see where the slop in a Detroit comes from. Zoom in on the center of this locker, the gaps between the square teeth on the axle gears and body are cause (not the r&p or splines). It’s inherent to the design.
By now it’s a moot point but… run it.
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Yes, there has to be some pre-determened clearance between the square cut teeth so that the locker can lock and un-lock to differentiate, or it would never unlock.

NO! its about the angle of the dangle.I know this but the subject was about the yoke slop and that is the main source.
What he said...Yes and turning the yoke till the r&p play PLUS the detroit play AND the axle shaft play in the side gears all get tight 1 direction or the other will give your input yoke a LOT of play over other units / designs .