What is the "same sound" that you are combating?
1.) A 'whine' at highway speeds that changes with throttle application?
2.)A clunk or groan that occurs at low speed in tight turns?
#1 is not affected by clutches. It could be somewhat affected by new bearings, but not likely. These diffs run hot;eventually the fluid gets cooked, the bearings then wear, misalignment occurs; gear tooth damage results; noise is created. Hence the "whine". New bearings might change the whine somewhat but won't eliminate it. The whine is created by the gear face damage which is permanent and no one MAKES any replacement gear sets for these. There is no actual fix other than to purchase a new differential carrier unit from GM (Rock Auto usually has a good price) for around $2K. Used diffs (as you've discovered) usually already have the damage and the whine. Now, if you could find a 10K mile pull-out...
#2. is a noise caused by the limited slip clutches 'sticking' and it is usually caused by the lube NOT containing the required friction modifier additive. (This happens often after a Cadillac DEALER replaces the lube with the wrong fluid). The fix is to replace the lube with lube which contains the magic additive OR buy the additive and ADD it to the existing lube. Again, as above, these diffs run hot, the fluid does NOT last a long time like it does in old live axles. It is VERY unlikely that the clutches need replacing. This noise (#2) isn't fixed with bearings, but simply by clean, proper fluid.