I was having a similar problem with our 96 Explorer. Did some research from this site and thought it might of been the LS differential. Week ago I unbuttoned the differential cover draining and replacing the fluid. (Make sure first that you can remove the fill plug!). I then refilled with Royal Purple 75w140. Did not add friction modifier since it is already included in the Royal Purple. I do have a tube of Lubegard friction modifier if I need to use it. Did some figure-eights in a parking lot and road tested. Still have the same problem as you described.

I felt like the rumble, like running over the small road turtles or a cattle guard for just a second or so, as somewhere beneath my feet.

Did some more research on this site and learned that the transmission is suspected and that Ford has suggested using the Ford friction modifier (now xl3) adding it to the transmission fluid. There are posts on this site where it has been added to Crown Vic fleet vehicles.

A few days ago I added 4 ounces of the xl3 to my transmission fluid. The problem seems to be gone now. I will know for sure next week when I take a 600 mile round trip over the mountains to the other side of the state.

I will find some of those posts and paste them here for you.

Casey