I can't really follow your description, but I know Snapper differentials quite well, having rebuilt over a hundred of them. There are 8 gears in there, the bull gear, smaller one on the sprocket that turns the bull gear, 4 small pinions, and the 2 axle gears. In nearly all cases like you describe, the axle gears are stripped. They are part of the axle. You will need a long and short axle kit. You may need other parts when you get it out and cleaned up, but those axle gears are the cause of your problem.