Assuming you have an NP231, the "Part-Time" will be on whether you are in 4Hi or 4Lo because in either case it is technically still part time, at least that is how my 91 is.
As for the lack of front axle engagement, my bet is a vacuum leak to the front axle disconnect which is a common problem and left me stuck in the snow twice. In a parking spot or a driveway, engage 4wd and move it enough that it should engage (5-10ft) then with proper precautions in place, climb under the Jeep and see if the front drive-shaft spins by hand. If it spins then it is probably your linkage, but if it does not then the problem is most likely in the axle, and as I said, most likely the disconnect due to a leak in the vacuum system.
Assuming it is a vacuum leak, I'm not sure the best way to actually find it because it really only takes a small leak to cause the problem. The easiest solution is to buy or build a Posi-Lok cable system and get rid of the vacuum disconnect all together.