Oh, man, that just doesn't sound good at all. I would get that car to the dealer with or without a loaner. There is only so much fluid in the differential and burnt fluid is never a good sign. It really has to be heated to a high degree to burn!
Holy shnikeys! + 1 on that! Just drop it off an take the bus. It'll be a lot cheaper in the long run. Sent from AutoGuide.com Free App