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I agree with Tom Endy. At step 6, if you do not have enough carrier pre-load you should remove one of the thick gaskets and replace it with a thinner gasket to increase the pre-load. You are on the right track, you just need to get the preload correct prior to moving to your step 7.
The "gears clicked" noise you describe could be normal. With a bit of clearance between the gears (backlash); when the pinion gear is rotated slightly from one direction to the other, you will feel and often hear the teeth initially making contact.
As Tom states in his Tech Article, you want the backlash to be as little as possible without teeth bottoming out. The contact pattern between the teeth should be centered between the top land (crest) and the base of the tooth. The pattern should also be somewhat centered on the ring gear between the heel (outer end) and the toe (inner end). If you are noting a contact pattern that is at the top land or crest of the tooth, you will have early failure of your gear set. Typically, with this pattern you will also have excessive backlash.
You have installed shims behind the carrier bearings so you have some ability to shift your carrier closer or further away from the pinion gear. By moving these shims from one side to the other you will be able to adjust the gear contact quite a bit. If after moving shims and gaskets you still have excessive backlash, you can install a shim between the pinion gear and the first (pressed on tapered bearing) this will move the pinion gear deeper into the assembly and lessen the backlash.
Setting up differentials correctly is not easy and it is lots of work. There is no easy way or shortcuts to do it. Good luck with your project.
Good Day!