I have a 1997 Dodge Grand Caravan with 251000 miles-yes- 251000 miles. For the past 6 or 7 years, it has had a horrible shaking while accelerating. It does not hop, but shakes side to side starting at 37 mph, thru about 50 mph. I have had it checked out at several different places, from transmission shops to actual Dodge dealerships. The answers never coincide with each other. The driveshafts have been replaced three times, wheels are true and balanced, tires are good, hub bearings changed, transmission mount replaced, and engine mounts checked. The only semi-plausible explanation I have gotten is that the differential bearings are worn in the case, and this is where the problem is. Also, there are no noises to confirm that diagnosis. I have owned this van since it was new, and I really like it, but would like it even better if I didn’t feel like a martini when I drive it.
I can’t comment on any possible issues related to the differential.
I will tell you that I have a '95 Caravan and have had chronic, similar problems that were eventually - mostly - solved by mount adjustments. As my shop more or less explains it, getting the front drive line aligned involves messing with the mounts, and the process is apparently more an art than a science. This was described to me as something fairly common to Caravans.
The other thing that can be out of whack (other than mounts or a bad axle joint) is the subframe itself.
The shop that I had best address this problem was a transmission shop.