Subaru Repair: transfer case, subaru impreza, case repair

Question
Hey, I just bought my very first car a month or two back (a 96 Subaru Impreza L Sedan AWD), and I'm having a few problems with it. In reverse there is a strong vibration seeming to emanate from the back, and also sometimes a rumbling sound comes from somewhere underneath the car while cornering. It also will bob up and down minutely with each revolution when coasting to a stop sign. I just read your response to someone with a similar problem (the cornering part anyway) with a Legacy. I will admit that I have four different brands of tires on my car, but they all seem to be the same size and wear and I doubt they are the main problem. The mechanic said it is probably my transfer case, and it would cost something over a grand to fix (which I'm not so excited to jump at, especially since its a annoyance more than anything). Should I try the GM limited slip additive? If so, where does it go? It is my understanding that my transfer case is connected to the back of my transmission and shares the transmission's ATF. Would the additive go right in there, or is my problem maybe in the rear diff, in which case, would I put some back in there with the 75w90 or whatever's in there? I'd appreciate your advice as I am a little unsure about what my problem(s) actually even is(are). Thanks.

Answer
Try the GM limited slip diff fluid first, and add it to the transmission fluid. Then drive in circles and / or figure eights for awhile. Regarding the tires, get 4 new ones. Sorry, but that really can still be causing problems. The best way to check them is to put the car on a rack, set the tire pressures all the same, and measure the circumference. If there is a 1/4 inch difference you may be in trouble. And your bobbing up and down could be a bad belt in one of the tires, or warped brake rotors. Make sure someone who really knows Subaru checks this one out good before doing an expensive transfer case repair, because, you may not need it! Good luck. Try back with more questions if you need too.