Silverado 4x4 pops outof gear 2w or4 w in backlash "NOT under power"

im going to say your clutches are going bad

There is no shims to reduce the backlash in a diff. The backlash comes from incorrect thrust washes behind the side gears . They wear and allow the tooth clearance to become excessive . Using new washes will bring the backlash back into specs.. Check bushes in the gears and shaft condition that the gears are mounted on.

Sounds like you didn't tighten it FAR enough to bring the ring and pinion gears to a match point. If it's making noise on the power up side of the gears..... I'll not explain, you may not understand. Anyway, lift the rear tires off ground and pull the driveshaft loose from the rear. You'll find there's play/slop in the yoke when you turn it back and forth. Lock the tires and tighten the nut until there's practically no play whatsoever. Play must be just barely discernable to have enough backlash in the ring and pinion, and since your not pulling the rear plate/cover to check backlash with a piece of paper between the gears, this is the only way to do it. And if it's been loose for a long time, even with correct backlash, it will still make a bit of noise until the teeth reset to proper wear position.

The issue is usually loaded gears. The secret is to try circles, one way and then the other. This preload is why 4x4 is supposed to be used on loose surfaces....The gears will disengauge after they unload.

chek the vac lines under the dash and under the hood

the front differential on those trucks are under built . have the backlash checked for an excessive amount

the gears in your rear end are probably chipped and broke or need shims

Most likely getting a little wear on the rear pinion, and you are experiencing the backlash from the gears. Put the vehicle in neutral (off) with the parking brake on and twist your drive shaft back and forth. There should be minimal play. If there is a bunch, you have a rearend rebuild in your near future.

there are adjusters on each side of carrier. do most of adjusting from driver side.need about 8 thou of backlash

If noise was not there before the gear change I would say that the gear are not set up correctly. Have the shop or person that did the gear swap recheck the gear backlash and the pattern on the teeth.
Too much heal or toe on the gear mesh will cause a decel whine or an accel whine. May just need to reset the tolerances on the gears.May be best to have a reputable shop check and reset the gear backlash and pattern. This is not much af a DIY type project unless you are experienced at doing gear work.

Hope this helps