Take a look at the electrical testing write up I did at the top of the page, get yourself a cheap multimeter, and start testing your circuit. I would start by looking at the switch and the shift solenoid on the t-case. As always though check your fuses first.

 

My father in law has similar problem . The other day he was backing out of the driveway , couldnt go far in the snow, so he put it in 4H while in reverse , I dont know if it ever engaged or not , right now '' service 4W '' light is on all the time , but the little green light doesnt come on when he puts it in 4WD . He wanted to know if it's something he can fix himself , and I couldnt give him straight answer . I dont see him climbing under that Durango , he's 63 and has bad knees .