Saginaw axles do that frequently. The problem started in the '70s and '80s when the axles were converted from a retained bearing to the "C" clip design. It is common enough that the aftermarket has "C" clips that are made in the standard 0.150" thickness, and in .005" increments up to 0.170".
In your case, it sounds like the side gears are worn/galled, and clips are not going to be a solution. You should be able to install new side gears and thrust washers, and a new spider pinion and not have to replace the carrier unless the side gears actually started to wear into the carrier itself. When you're all finished, longitudal lash in the axles should be as follows:
Maximum Allowable Axial Play in Axle Shafts
Code:
Standard Differentiual Limited Slip Differential
Axle under 45,000 service miles:
0.040" 0.065"
Axle over 45,000 service miles:
0.055" 0.085"
If you hae to replace the carrier, you'll have to perform the complete setup procedure for the axle, including setting pinion depth, centering the carrier, preloading the bearings, and checking contact patterns. If you're not up for that, the replacement axle idea may be a good alternative.
It's fairly easy to check: