You might check the pinion pre-load and drive line slack before tearing into the rear end. It is quite a job to correctly set the mesh on the differential gears.
Block the wheels, pull the parking brake and put the transmission in neutral. Rotate the drive shaft back and forth while watching to see if the pinion flange moves in and out. If it does, drop the rear u-joint and carefully tighten the large pinion nut just enough to remove MOST of the in/out movement. DO NOT OVER TIGHTEN THE PINION NUT. If you over tighten the nut you will have to disassemble the rear end to replace the crush sleeve (if you can find one). The necessary tightening, if needed, is most likely less than 1/4 turn of the nut--The idea is to take out MOST of the in/out movement without causing a bind in the pinion.
Refer to the 1949-50-51 Ford Passenger Car Shop Manual (green bible) Pages 103-111. Note the preload is 13-18 INCH pounds or 15-20 INCH pounds on the late model rear end. You probably have a late model in the 1951 vehicle. Page 107 left column explains the procedure. You will need to raise both the rear wheels and release the parking brake to properly set the preload.