Do you have a pic of the pieces you found in the pan? Without seeing them it could be anything. Differential failures arent real common but they do happen time to time regardless of driving style, though burnouts with non HD differentials put them at risk for failure sooner than others. Often if the diff pin is broken and the cross pin and gears are bad and the cross pin has been spinning in the diff case you will need to replace the whole assembly. Most places wont sell just the differential gears and cross pin so you will need to buy a complete one. You should also check the axles to make sure they arent bad as well as suspension compenents as this can also cause vibrations during a turn. If the metal you found in the pan is non magnetic then it is likely aluminum from the cross pin in the diff grinding away at the inside of the diff cover and you wont really know what you need until you pull the cover off and take a peek inside.
