There should have been a lot of whining from the diff long before it got that bad. Rebuilding the diff is not a DIY job and the gears are likely to be noisy after even if they don't look damaged. I don't think it's all that common for the diff to go out so there should be no problem getting one from a wrecker.

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)