I think I need to replace the rear axel in my.

That's a major job and seldom necessary. Before you launch off on a wild goose chase, there is a tool you might be able to borrow or rent from an auto parts store that borrows them called the "Chassis Ear". It is a set of six microphones, a switch box, and headphones. You clip the microphones to suspect points, then drive around while listening with the headphones. You can move the microphones around to zero in on the source of the noise. Be aware that many mechanics have never seen or even heard of this tool. Suspension and alignment mechanics use it to find rattles, squeaks, and other noises.

Some limited slip differentials will make a chattering noise when turning. That is eliminated by adding a small bottle of an additive containing whale oil. The dealer's parts department has that.