My truck isn't stock, and I've had it long enough to try lots of things.
Driveline vibrations certainly are speed specific and probably what you are dealing with. I doubt your diff setup is bad since you drove it so far already.
Step 1 verify driveshaft phasing. The Y parts of the yolks should match. So the ones welded to the driveshaft and slip yolk should be exactly the same. I have a feeling your FRONT problem is driveshaft phasing going by how severe it seems. It is easy to be off a tooth on the driveshaft and it "looks" right until you put the shaft on the ground.
While you have it down on the ground, go have it balanced. maybe you knocked off a weight on a rock?
Still working on the front...The only thing that can be wrong with the t-case is wobbly output bearings at the t-case. I kinda doubt this because if the bearings are this beat, they leak like the Exxon Valdez. EZ to check. just lever on them up and down. On my mini truck, it is easy to change only the front output.
Uncle Ben already described the angles thing. Basically, with single u-joints on each end, the flanges need to be parallel for smooth operation. Remember, with leaf springs there is some axle wrap under "normal" driving so you tend to "pre-tilt" the rear down about 1/2 to a degree or so. Oon the front, caster is WAY more important for day-to-day use.
Get the caster where you want it first. We'll fix your driveline in a second.
Next up, the "double cardan" you've seen 'em. 2 u-joints back-to-back. UB described these too. Forget the physics arguments. Here is how it works. Point the rear diff flange directly at the t-case flange. Put the double cardan at the t-case end. Single U is all you need at the bottom.
OK. How do you decide which way to shim? Down for parallel? or up straight at the t-case flange for a double cardan? Well, which way is closer? Can you afford to have a double cardan driveshaft built? Want to get the nose of your diff out of the rocks and the driveshaft angles reduced? Don't care that the fill hole is now useless on the diff and the anal retentive will tell you your pinion bearing won't get lubedt? shim the nose UP and get the double cardan shaft under there.
Decision made? Uncle Ben also described how shims are frowned upon. Yeah. U-bolts end up too short on 1 side, e-brake cable runs into the leaves, shims split and spit out....sucks. Here is whatcha do. Do the math as best you can and buy shims as close as you can calculate and estimate. get steel shims only. install and TEST. when you find a shim you like, weld them to your perches. the BETTER OPTION: drop the axle. cut off the original spring hangers and weld on new ones. Rotate the axle housing to the new, optimized and tested angle. now when you bolt it up everything fits right and, "Stock". Cutting and welding hangers isn't a big deal really and the result is much better than an angled shim.
On the front: Run the double cardan. even if you have non-parallel flanges, the vibration won't be as bad.