The differentials in those trucks are almost bullet proof, however it is not unusual for them to be noisy. It could also be the driveshaft center support bearing. It will cause a vibration that could emulate transmission and/or differential problems. Be sure to mark the sections before you take the driveshaft apart.
Fred


1956 3100 Pickup/Red/350/3sp OD/PS/Disc Brakes
1957 Bel Air Sport Coupe/Red/355/TH350/PS/Disc Brakes
2017 Silverado LT Single Cab SB/Black/5.3/6 Speed Trans
1947 Willys CJ2A w/F-Head engine