Adjustment
You have to be VERY careful adjusting the gear. You don't want to go any tighter that zero backlash or you will destroy the bearings in a hurry. 1/4 turn is a huge adjustment! To do it properly, you need to pull the gear from the truck. Turn the shaft back and forth to pump out all the oil. Put the gear in the vice and then put a vicegrip on each shaft. Holding the output, slowly move the input back and forth feeling for the gears to touch. Adjust in small increments until there is no backlash. Its similar to setting up diff gears except you go for zero backlash and have an adjuster instead of shims.
I agree that the problem is probably elsewhere. If the box is loose enough that adjustment makes a difference, it's probably shot anyway. Also the output shaft will have a little side to side movement from bearing clearance. It should not go in and out at all.
Good luck,
Dave