The system is very simple. Cable housing goes from the handle to a bracket on the left side of the t-case, cable extends to a bracket that connects to the selector lever on the t-case.
1. Check the fuel prime pump, could be air in the fuel.
2. Isolate if it is the t-case causing the issue, or the handle/cable/linkage bits.
The linkage can easily be soaked with water and then freeze, I would get under there and check if it's caked with snow/ice. If it is, chip/melt away what you can, and hit it with some white lithium grease and try again,
If it's clear disconnect the selector or cable (can't remember which is easier) from the t-case and try to move the bits via the lever. If the cable is free, there is something going on in/on the t-case. If the lever binds, there is something in the lever/cable causing the issue.
If it's the t-case, most likely the cogs/chain in the t-case are not meshing when you are trying to shift. When the lever seems "jammed" again, shift trans into D/R, you will feel the engine load, shift back into N, and try again. If that doesn't work shift per owner's manual (rolling slow, shift to N, switch t-case selection, reengage trans, drive off).
If the t-case checks out, hit the linkage with some white lithium grease for good measure, and start checking out the condition of the cable. Engage the t-case in 4lo, and slide back under, check for fraying/debris/ice/gunk on the exposed part of the cable. Have someone in the truck move the lever back and forth (2wd-4lo) to see if it's binding in the cable housing. Clean as needed and hose down with, you guessed it, white lithium grease while someone works the snot into the housing.
White lithium will expel water, not wash off, and not freeze.
I would say I have been in some high-risk environments for this situation, never once had an issue with freezing of the lever/linkage under there, but once in a blue moon I need to re-sync the guts of my t-case.
It was 4* when we broke through that creek
