I would check u-joints first but really doesn't read like u-joint issue.
Then I would drain differential and look at the oil that comes out, obviously metal filings would be a bad sign, then pull the cover and have a look at the ring gear.
Rebuilding is not too bad of a job it does require some specific knowledge on shimming for best fit but there are many "how-tos" and videos on that.
Or you can try to find a Ranger axle similar to yours, maybe even with limited slip and swap the whole axle.
Good read here on the 7.5": Ford Ranger/Bronco II 7.5-Inch Axle
If looking at wrecking yards the door tags can help, read here: http://www.therangerstation.com/tech...le_codes.shtml
Long shot would be rear brake issue, there is some play on the axles when turning which can make grinding issue louder on the side with brake drum issue