You need to post some more close up and detailed photos of the suspension, front and rear, so we can dissect this thing!
Chances are the rear end is walking when flexed because it looks like the suspension links are stock length. They are pretty short and with that much lift they are going to be very far from the horizontal angle they would be if not lifted. The more angle to them, the more dramatic the change in wheelbase as the suspension cycles.
The aluminum diff can be replaced with a steel one out a V6/manual trans GV or XL7. Or a solid front axle.
The stock t-case can be upgraded with low-range gears from TrailTough or Calmini. I think 4.24:1 is the only option, but it will improve the crawl-ability of the truck by a hundred-fold. There are other options that involve using a different tranny/transfer case combo, like doubling up a 2wd Tracker tranny to a Sammy t-case which has more options for gearing and also has high-range reduction, or doubling up a Tracker t-case to a Toyota t-case, etc, or installing an underdrive box between the tranny and t-case, etc. The trickiest thing with these types of setups is the overall length of the package and the short wheelbase of the vehicle can lead to driveline vibrations and other related complications. But you're a Jeep guy so you probably already know all about that.
I'm sure it wouldn't be too hard to find a set of softer springs for those coilovers online. At least you know what's in there now so you have a baseline to tune from. The big question is, do you want to spend money "perfecting" that IFS or just do a straight axle swap and have something reliable if somewhat less side-by-side-like in high-speed performance?