Sounds like it was old AND wet. It's normal for the gear lube to stink.
(Aside: The only worse smell from a car than old gear lube is GM factory supercharger oil. I don't know what that stuff is, but is it animal-based. You would rather smell your butt...)
If the vibration came on recently, it may or may not be from the contaminated gear lube. If it is in the rear axle, chances are you've wasted a bearing. This may not be so bad, you'll just have to check it out. Did anything come out with the gear oil? Any chips or visible problems inside? When was the differential lube last changed? You will need to examine and clean or replace the wheel bearings. You should also look at the carrier bearings. You will probably have to pop the carrier out to do this. If so, make good notes of how everything came apart. As long as you put it back together the same way, it will be fine. If you take the carrier out, take the opportunity to check out the pinion bearings.
I think you should still do a full check out of the driveline. Look for bad U-joints in the driveshaft, loose slip-splines and check to be sure the pinion nut is tight.
Since there was contamination, you should check out the axle and pinion seals and check the vent tube for cracks/leaks.