Transmission Tailstock Bearing Play

jpikaart said
The transmission shops says that the bearings in the differential may need replacing, would this be the cause of the vibration at 45 MPH and higher?

It very well could be. Since you say it's more like a hum now I'd really suspect differential bearings. Plus you should never have to tighten the pinion nut yoke. If it was loose that's another sign the bearings may be toast.