trucks have a limited slip rear end to help in turning. if both back tires are in the air they will turn,the drive shaft wont turn because its in park. to check the vibration you might check the balance of your tires
There is no shims to reduce the backlash in a diff. The backlash comes from incorrect thrust washes behind the side gears . They wear and allow the tooth clearance to become excessive . Using new washes will bring the backlash back into specs.. Check bushes in the gears and shaft condition that the gears are mounted on.