You definitely want oil in it. If you haven't already, check your front/rear diff as well. They most likely only have grease as well. Fill the center with something like 7000wt diff oil. The front should have something in the neighborhood of 5000 and the rear should have something around 5000 or maybe 3000 wt. I would think the owner's manual would have a suggestion on what oil to start with.

As it stands right now, I'm guessing you're front tires probably balloon up under heavy acceleration and the rear end is loose. The diff oil will restrict the differential action and send more power to the rear wheels which will improve acceleration and help keep that rear end in line.