Your car came stock with a Trac-loc limited slip differential. What has happens is that after 20 years and I'm sure 150k+ miles is that the clutch packs and discs in the diff wear out, and the effectiveness of the limited slip are greatly reduced.
All you have to do is have the Trac-loc unit rebuilt. A rebuild kit will run you about $100-$120. If would also be a good idea to have bearings in the axle replaced as well. If you were considering changing out the rear gears, this would be an ideal time to do it.
Well you should not be doing burn outs as it puts a tremendious amount of strain on your drivetrain. But if you feel that you must you can't go wrong with 4:10 gears but I would only go with 50% locking so you don't lose your normal drivability. If you are not looking to burn up a lot more gas with such high gears you could go with a lower geared limited slip differential.
You didn't mention if this was a 4cyl, V6 or V8. I don't remember if the fox body GTs came with something other than the 302 V8 or not. If it's the V8, it will already have a limited slip differential and it's just worn out. If it's anything but a V8, it will have the anemic Ford 7.5" rear axle. If that's the case, you will want to get a different axle assembly all together because the 7.5" is a very weak axle.
You need to get the complete rear axle out of the same generation of mustangs which has a B series number. These are the limited slip units. Now more then likely these will be a five lug unit, so get the rest of the part needed for the front, or check into how to make it into a four lug rear end. The reason I suggest this method is because to just put a LS diff in the present housing might prove to be more expensive to do and it may not stand up to what you are very long.