Ok so I took a closer look at all the photos, one thing I see that might be your culprit... Every single 2/3 piece driveline ive seen on these model Ford's, the first shaft at the t case is always parallel to the body, then at the carrier bearing joint that is where it angles down to the rear diff.
That huge body spacer at the carrier, does it come out? Looks like it needs to be raised a good 4"
Crawled underneath today to take a look and that bracket looks like it's got about the same amount of rust as everything around it, and is riveted to the cab. I expect that Ford did actually put it there. I could potentially shorten or remove it, I suppose. Except...
Been diving through other forum's existing posts and found one where someone realized their motor mounts were the culprit. While I was down looking at the bracket, I took at look at my mounts. Both of the motor mounts appear to be doing their jobs, and look pretty intact. The mount at the back of the transmission, one of the bolts going up into the transfer case was a few threads undone (which I was aware of from previous searching), and the other was completely missing (which I was not aware of).
I put my phone up to record a video holding the brakes and trying to accelerate. I have a feeling this motion is my main problem. I don't expect there's supposed to be this much twist at the transfer case.
Headphone users beware, loud IDI noises (Won't let me upload video here, so here's a youtube link)
Video Before tightening bolts from transmission mount into transmission
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I went and pulled the one bolt that was still there to check the threads on the other side, make sure someone hadn't stripped it or crossthreaded it before, and luckily found the missing bolt resting in the channel of the transmission mount. It looked as if it has been just laying there for probably much longer than I've owned the truck. I tightened both bolts that went up into the transmission, but did not try to tighten the two nuts that came out the bottom of the crossmember, which in hindsight, I should have.
I took another video at a slightly different angle after actually attaching the transmission to the transmission mount. I'm not 100% sure it's any better yet, but In the corner of this video, i think I can see the whole transmission mount moving. I'm done for today, but I'm hopeful that making sure the transmission mount is securely mounted to the crossmember in the morning will result in much less twisting and moving around.
Video After tightening bolts from transmission mount into transmission
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Even if this isn't the end cause of the problem, it's still nice to find something that is actually wrong along the way.