I am going through this on my '96. It is the tires. You have out-of-round tires. This was caused by a worn out front suspension.
I had a set of Coopers that got to the same scary, out-of-control point you describe. I had the wheels and tires checked for roundness and balance. The front ones would no longer balance at all and were egg-shaped. So I had TWO different frame and alignment shops inspect and align my front suspension. Both shops pronounced everything to be in fine shape. This really surprised me since they usually want to sell and install everything whether you really need it or not. (I put in a new track bar anyway after the first aligment and went somewhere else to have it all done again. )
Deciding all was now well, and that it had been an alignment, track bar, and poor quality tire issue, I went back to my favorite (and expensive) BFG Mud Terrain tires feeling confident the front suspension was OK.
It isn't. And now my expensive front BFG's are trashed, cupped, howling, and shaking just like the Coopers. In fact, if you roll along very slowly, it feels as though the tires are actually square. The thunk, thunk, thunk of each lug turns into whomp, whomp, whomp and then into a banshee howl as your speed increases. Just the fronts; the rears are fine and quiet.
Those two front tires are far louder than all 18 tires on my semi combined! My wife and kids can actually hear my tires when I'm still 1/2 mile from home. My 250 watt stereo must've got so depressed at being drowned-out that it quit working, too.
I hate this crappy Dodge front suspension. How the heck can you ruin a Dana 60 so badly? Ball joints, central axle disconnect, coil springs, integral wheel bearings, track bars, etc. is how. Dodge did all they could and did a fine job of turning treasure into trash with their rotten front suspension.
You need your suspension rebuilt and THEN you need new front tires. And, apparently, the Dodge is good at disguising it's frontend woes, so don't believe any shop that says your frontend is good.