So I wanted to let you know whats worked for me.
About a month ago I blew out my rear diff/driveshaft. I got a new rear diff from east coast gear and a local driveshaft.
Fast forward got the truck back running in 2wd. I tried out the 4wd to make sure everything was working properly.
When i first tried the 4wd I could not get it to engaged no matter what I would do. Similiar to your situation. What i did was put the transfer case into neutral with the 4wd button not pressed (2wd), make sure all the lights are off on the dash. This puts the transfer case and the electronics on the same page. What i found was that my transfer case once in neutral the 4wd lights were engaged but my button wasnt. I pressed the button and put transfer case back into 4hi. The truck drove normal, i disengaged the 4wd and it clunked out. After I got everything on the same page, I still have problems but I know it was engaging.
I have noticed since I swapped my driveshaft and diff that I can no longer swap to 4hi on the fly. I have tried rolling around 15 mph and i get a lot of binding and grinding. Sometimes i even get some tire chirp. I immediately stop, switch between neutral, reverse, and drive a few times, and drive a few hundred feet and i can almost everytime get it to swap into 4hi. The truck drives fine once it is finally engaged but I noticed if i do not let it engaged i get alot of what i think is binding. The truck almost feels like its pushing the front wheels. I thought I might have a problem with my transfer case but it does engage when I stop and switch between neutral and drive a few times.
I think i will swap the fluids and see if that helps with engaging on the fly. I used to be able to swap going 40mph down the road, I did it on multiple occasions without any clunk and chirp at all, I did not even hear any grinding/binding. I think theres a chance I am low on fluid in the transfer case thats why the truck has a hard time engaging at speed.
Sorry for the long post and I hope it makes some sense. In my case everything was out of wack.
best of luck and hopefully you get it working soon, winter is coming!