A Limmited Slip Differential Question

This question isn't about my Duster, it has to do with my 1998 Dodge Ram 2500. As I've always understood it, with one of these differentials the power is supposed to transfer to the wheel with better traction. My question is, how quickly should that happen? My driveway is fairly steep. At the top by the garage there's room for two vehicles side by side, but the actual driveway is only about one and a half full size cars wide. Several times now when coming up the driveway my truck has become stuck when the right rear tire left the pavement and got on the grass while going around another car. When this happens only the right rear wheel spins. The wheel on the pavement just sits there. This is exactly how a standard rear end behaves. What gives? I ordered this truck from the factory and paid for a limited slip differential. When I run the vin code it says I have one, but it sure doesn't act like it, at least not the way I always thought it would. Am I expecting too much? Does it take more time for the power to transfer to the left wheel than I am giving it? It's a Dana 80 rear axle. Any thoughts?
Outta my league, but is it 4wd? Can ya creep up the hill? I Have had several trucks that were supposed to act as expected, and it the same crap. Tire w/ less traction spins...