From the movement you have, I'm assuming you don't have conrad bearings (Green bearings), but have the stock tapered roller bearings.

I don't see the differential thrust block in your video. And, that side gear movement is normal. When under torque loads, the elements in the TruTrac 'find their own center' and the pinion gear loads keep the side gear centered.

Dr. Diff's site tells you that you need to install the thrust block when using stock adjustable bearings.
http://www.doctordiff.com/8-3-4-eaton-truetrac-differential.html

The thrust block should have been packed with your TruTrac. It looks like this:


Once you get the thrust block in, you'll be good to go. If you didn't get it, call Eaton. They have pretty good customer service, and they'll get you one.

Bob

TruTrac thrust Block.JPG
AAR 4-speed 3.91, Tor-Red;
70 440 6 pack Roadrunner 4-speed 3.54, Plum Crazy;
68 Formula S conv 383 4-speed 3.23, Electric Blue;
69 Barracuda conv Slant 6 OD4 2.94, 71 B5 Blue;
78 Lil' Red Truck, Red;
70 Challenger S/E. 505 6 pack, Passon 5-speed, 3.55, B7 Blue