My money is on the pinion seal is leaking. The oil/lube on the carrier is from it being blown back there when your driving. Notice it is wet before the carrier gasket.
You'll need to correct pinion seal, based on which carrier you have. The actual fix is easy, BUT you better have either a lift to get good mechanical advantage breaking that bolt loose to remove the yoke, or have an impact gun. When finished, the torque on that nut is 250 ft/lbs or greater.
