Shudder suspected from rear differential in 2004 Ford F150

My 2004 Ford F150 only has 20,000 miles on it but makes a shudder from stop turning left or right. I asked the dealership about it and was told that it was likely the clutch pack in the rear differential sticking. I was told I could try changing out the fluid while also adding a few ounces of some special lubricant. But, they said it wasn’t a guaranteed fix. Apparently, they have tried this fluid change and drive the vehicle in figure 8’s trying to get the lubricant between the plates. They also said that this was a known problem to Ford. Lastly, I was told it wouldn’t hurt anything to keep driving it. I checked some other online forums for Ford F150 owners and this does seem to be a recurring problem. I contacted Ford’s customer satisfaction folks and was told of course that this was news to them. Any advice you can provide is appreciated. I’m thinking about having the clutch pack replaced but obviously hate to spend the money if I don’t have to. Thank you!

04…new body style or old ? ( does it look like an 03 or 05 ? )
Has the rear axle oil been changed recently ?

New body style shows a TSB # 06-04-04 which says to replace the clutch set.

Hi Ken, Thanks for the response. My truck is a new body style 04 with only 20,000 miles on it. The oil in the rear differential has never been changed.
What does “TSB” stand for?
Thank you for the help.

It’s Technical Service Bulletin.