Dodge 4x4 Doesn't Engage

Remove the actuator motor assembly from the transfer case and try to manually engage the transfer case, if it is easy to engage then the actuator is the problem.

check that free wheeling hubs are engaged (manual or vacuum operated or electrical operated )

Your transfer could be bad.

Check the vacuum lines from the engine to the transfer case and from the transfer case to the actuator on the front axle.

4x4 issue could be either the TCCM Transfer Case Control Module, or the transfer case shift motor.

When you turn your dash switch from 2wd to 4HI, can you hear the shift motor underneath operating?

This could be defective 4X4 actuator motor or jammed transfer case. Has the fluid ever been changed in the transfer case? if not that can cause the case to jam and not engage.

the front axles have pinon gear box that the shaft from the transfer case goes to, it has a vacuum hose to it or a solenoid actuator check these for proper function this is how the power is turned on to the wheels.

actually that first soultion is wrong it is correct on smaller pickups but on the full size pick ups they use a differnet sytem
there a electric actuator on front axle
there be two elctrical plugs on front axle u want the bigger of the two other one is just to turn the 4 wheel drive light on
un plug actuator then remove actuator after it removed plug it back in but leave it out of axle turn key on and shift down to 4 wheel drive the actuator should move out if it dont that your problem if i does move this isnt your problem it either the tranfer case or in front axle

shift into 4wd and move truck a couple of feet.
if light is still off, park truck block the wheels set the park brake jack right front wheel and try to hand spin the wheel if wheel doesn't turn 4x4 is functioning properly but there is a wiring problem use an ohmmeter to check the 4x4 switch wiring fuse and bulb
if wheel turns freely the front axle or transfer case is not engaged
attempt to hand spin the front driveshft if the driveshaft will spin the transfer case is not engaged check the transfer case shift linkage, problem is in linkage or transfer case internals
if the driveshaft will not spin the transfer case is engaged properly
with trnsmission in neutral or park start engine measure vacuum on the two vac lines at front axle shift motor . one line should have approx 20" vacuum. other line should have none
if no vacuum measured look for vac leak or bad vac switch
if both lines have vac look for a bad vac swithc or ruptured cad actuator diaphragm
if one line has 20" +- VAC
move transfer case lever to 2wd measure vac again lines should have swapped vac signals
if no vac switch bad or 4wd shift likage needs adjusting
if yes vac motor bad or stuck or shift fork stuck
remove vac shift motor from the axle attach vac lines to shift motor engage disengage 4wd
does shift motor plunger move at least 1/2 inch
no shift motor bad
yes check axle shift fork and vac motor for binging

On the axle there is a vacuum Actuator that engages a collar on the axle that engages the four wheel drive if the axel splines are worn or havent been lined up they will get burred and won't allow the collar to line up. You can remove the actuator and see the splines and determine if they need to be fixed.