Yes, check the code for which rear-end you got. It should be stamped on the door tag. If you don't have traction-lok, a new differential housing will cost about $225. If you do have traction-lok, you can buy a kit for about $70 to rebuild it. As suggested, you may as well put gears in while you do this.
Unit 5302, that's really bizarre that a GT would come without traction-lok. I have an 89 GT convertible and it definitely has it. Did your uncle buy the car new? If not, someone probably put in a new rear-end that didn't have traction-lok. What does the door jamb sticker say he has? It should have an:
8 - 2.73
M - 2.73 w/limited slip
Y - 3.08
Z - 3.08 w/limited slip
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351W 89 Mustang GT Convertible