If something is hot enough to vaporize/burn gear oil, it will be HOT to the touch. Take a ground cloth with you, drive until it smells strong, crawl under, and feel the diffs, tcase, tranny, axles...

Most gear oils have castor oil in them. Have you ever smelled straight castor oil? Note that breathing vaporized castor oil fumes will give you the galloping trots...

Not responsible for advice not taken...