Ford F-150 - 1997-2003 :: 1999 V6 4.2L - Rear End Shudder / Vibration

I have a 1999 F-150, 4.2L V6 with 206,000 miles that recently developed a rear-end shudder/vibration. According to my door jam axle code (27) and the axle tag (3 37), I have what appears to be a conventional rear differential, even though most of my research would indicate this rear-end is for the F-250. In any case, and based on other online research, I changed the differential oil with a 75W-140 synthetic and added 4oz of a friction modifier. Now the vibration appears to be worse. It primarily vibrates when taking off from a stop, whether turning or not.

I have felt the vibration at speeds up to 60mph, but it is more noticeable at lower speeds. The vibration also seems to diminish after driving the truck for longer periods of time, particularly if at highway speeds. In other words, when I drive to and from work, which is about 30 minutes each way of mostly highway, the vibration is barely noticeable once I exit the highway and slow back down to 35-40mph. When feeling the vibration, if I decelerate, the vibration immediately ceases, but starts again when I accelerate. It's not as noticeable if I accelerate harder, but I generally don't do that. The gears inside the differential didn't have any noticeable wear, chips or slop/play. What may be plaguing my usual trouble-free F150?