I'm not sure if this was mentioned before (forgive me if so, I didn't re-read all the way through). It is also very basic, which is sometimes where we forget to start.
Twin stick conversions that still have the safety "pill" / "pin" between the shifter rails internally need to be operated as follows:
The front axle shifter must be in Neutral to shift the rear stick between it's 3 positions. Once the rear stick is in L or H, the front stick can then be shifted between N and the same position the rear stick is in.
(the rear will not move from its L or H position until the front is placed in N position)
That is to ensure the front is not in low, and rear in high, which will damage your t-case.
Now if that's all good, and you still can't shift it, here's what I would do.
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Start by locking in the front hubs (so when you move the tires, the front drive line will move the t-case gears) - jack up the vehicle (front and rear on Jack stands) and have a buddy over - give him beer (or cool-aid if too young for beer
)
Then remove the linkage from the shifters to the rails (just have blank rails up front)
Remove the caps from the rear of the T-Case carefully if possible without damaging. They just press in, so twist back and forth and pull for a while may work, or may have to destroy them and tap in new ones after completed.
Grease the front side of the rails near the seal / case and the rear the same (should / may see some of the rail protruding slightly from the rear)
Work on getting the front axle shifted to N position. - Once there (N) move on.
You may need to have a buddy spin the tires up front or spin the driveshaft manually so the gears in transfer case move slightly - that was the purpose for the buddy... not to just give away your beer to... 
Work the rear axle rail between all 3 positions while re-greasing if necessary multiple times, until the operation is smoother.
Once rear shifts smoother, shift it to L - then shift the front between L and N repeatably until smoother (greasing as necessary) - re place shift rail for the front back into N
Shift the rear rail into H - then shift the front rail between H and N repeatably, until the operation is smoother (greasing as necessary)
Then put the front back in N and Rear in H. Replace the caps on rear of rails (old or new) and lower jeep back to the ground & Re install shifters to the shift rails - Take it for a small drive in H 2WD then work on playing with the shifting between all locations. (remember your hubs are still locked in 4wd up front - so no freeway yet.)
If this does not work, I would by a rebuild kit from Novak (or others, but I liked Novak's) and find a buddy with small shop press (i bought one, so much easier to work on some stuff) and go to town rebuilding. Mine shifts world better after a rebuild
I promise you will come out of this issue knowing way more than you do now about the transfer case
