Front or rear differential? That sucks man! I don't know anything about the R's differentials, all I can say is that I feel your pain. I had the same thing happen to my 2001 Audi TT. Prev owners toasted the rear diff and I didn't notice till after I bought it and drove in snow a few months later.
Similar new cost of over $4k CAD for a rear diff. I got one for $450 used off an auto wrecker locally. Paid a shop to rip out old, install used - didn't work! The auto wrecker THEN swapped that one for a NEWER used one, I had to pay to rip out the other and install this one. Twice the labour. At least that one worked.
Beware that wreckers will sell you their oldest part they have.