www.ringpinion.com
ok.... going back and trying to sort this out..
first, there is a passenger car 12 bolt and a truck 12 bolt axle. If your posi came from a truck 4.11 then it should fit your 3.73 gears. The carrier break is 3.42/3.73. If it came from a car, sell it and put your non posi back in.
2nd, I don't see anything about contact pattern in anything you posted so far, so...
Assuming you have a truck posi, Here is my advice..
Remove your ring gear from your carrier. This will prevent contact interference between ring and pinion while establishing carrier pre-load.
Set your carrier up with enough shims so that you have 0 end play. For the garage install, your carrier should go in "snug" since you aren't using a spreader, but should turn freely once installed and caps torqued.
Once you have your total shim pack thickness established for 0 end play, you can remove the carrier and re-install the ring gear.
Now, I recommend recording your total shim thickness for reference. You can now establish your CONTACT PATTERN. judging from what you are saying, I would start with all the shims on the passenger side of the carrier. If you have questions about which way to move the ring gear, I can get into that if you need.
In reference to you talking about the axle shafts moving right or left, you could shim your carrier all the way to one side and it wont move your axle shafts any significant amount. you do need to ensure that the spline count is right between your shaft and the side gears.
One more thing that I highly advise is inspect or replace your axle bearings, and replace the axle seals no matter what.
Another thing you can do while you are into your axle is check pinion pre-load. You don't want to allow a failure to ruin all your hard work!