Oil....... What is good to use?

If you find a GL-5 spec 80W-90 with limited slip additive already in it, you can use it for both your trans and differential. It will say for use with limited slip differentials right on the container. If you get some rear end chattering, you can add a little more of the lsd additive. It shouldn't take much with a healthy diff. You don't want to put in too much.
Brand isn't especially important. The stuff you need isn't fancy, so any brand will do. A quick search turned up Castrol, Kendall, and Valvoline with LSD additive readily available local to me. If price is a concern, check out the gallon of Super Tech at Walmart. Super Tech tests very well, but if you stick with the major brands, you can't go wrong.