I have this EXACT same problem with my '85 Brat. I describe it as a "wobble" ... only when accelerating in first gear, not as bad when in 4WD, under my feet or under my butt... I've searched this forum from one end to the other and tried anything that even seemed remotely similar...this place is great!
Mine started when I had the clutch and front axles replaced. The shop also replaced the transmission mounts. I took it back to them several times and they acted like they could not feel it. But, I have driven this car for many years and the wobble is most certainly there.
I've recently replaced ALL of the axles on the Brat...the front ones twice thinking the problem must have been with one of the new ones.
In my quest to figure this out I have replaced both U-joints and the differential mount, but the wobble remains. Replacing the diff mount has helped with a little road noise, but not the wobble. I don't believe the rubber being gone out of your old diff mount would allow the thing to "thrash around". You still have the weight of the car on it...only it's metal on metal...I drove my Brat for many miles without that bushing...
I've tried adjusting the stablizer (dog bone) and it seems to have made a slight difference. I have purchased new motor mounts and a new bushing for the stablizer but I have not installed them yet... I really believe/hope it is the motor mounts.
As for the cause...I feel like wobble has been there for a while but not very apparent since motor and trans mounts were bad, it was floating...when the shop replaced the transmission mounts it tightened up the drive line and caused this wobble to be more apparent.
I'll be installing the new mounts soon...I still drive the Brat every day with the wobble. I will make another post after installing the mounts.