Good luck with the transfer case swap. It really isn't very difficult if you can do it indoors without freezing. Certainly not worth paying someone $1600 to fix it for you.
I've had mine apart a few times...
First, drain fluid. Then remove the front driveshaft, and disconnect the rear driveshaft from the transfer case (TC). You can leave the rear shaft attached at the rear diff. I usually strap it to the exhaust to keep it out of the way. Next, throw a floor jack or trans jack under the trans and remove the trans mount crossmember. Disconnect the TC vent tube and wires. Remove the TC-to-trans bolts that you can get to. Lower the trans a little to get access to the top bolts, then support the TC and remove that last bolt or two. Pull the TC rearward to seperate. Some fluid will pour out from between trans and TC, so have rags ready. Then remove TC from under truck. It only weighs 40lbs or so, so you should be able to lift it out by hand. Use another floor jack to support if unsure. Replacement is the reverse of removal. Make sure you replace the gasket between the trans and transfer case.
*The exhaust cat-back may have to be disconnected from the cat pipe if it is in the way or keeps the transfer case from lowering.
**You might have to jack up the truck and put it on stands if you can't fit under there. Mine's high enough that I can leave all 4 tires on the ground. I just hop on a creeper and roll under.