Try putting a ramp under the passenger side of the Eagle and leave the driver side on the ground. No fluid will come out as you pull the cover. You simply remove the four bolts, pry gently to pop free of the old gasket material, clean up the gasket surfaces, put a hose clamp on the shaft of the shifter, and then reinstall it with RTV gasket maker. There are two little bushings at the tip of the fork that you should try to keep from popping off and losing. When you put the cover back on make sure the fork slides into the groove of the coupler and that it is still there with the coupler all the way outboard as you finish tightening the bolts.
I drive my Eagles as All Wheel Drive vehicles. My NP129 versions have rope replacing the shift mechanism to keep them in AWD permaneantly, identical to the 119. The 2WD isn't at all necessary in these cars. The nice thing about the 129 is the possibility of mechanical failure. If I need to tow on a tow dolly I just crawl under and pop it into 2WD. If the front axle needs to be removed temporarily I just put in outer halves of CV shafts to keep the wheel bearings intact and then put it into 2WD.
As far as my choice of transfer case, the only one I'm installing in my Eagles is the NP242. I use the original driveshaft with a piece of pipe installed to limit the travel of the original slip joint. I have pictures of this in my project thread.
The 242 has 2WD for towing or repair situations. It has an open differential perfect for driving the Eagle in AWD. It has a fully locked position for totally snow covered or otherwise extreme conditions. It has a low range perfect for slow moving utility.
The 242 does not have a viscous coupler. Don't fear this. Subaru all wheel drive and just about every other AWD vehicle are open differential transfer cases, including 1986 Eagles with the 128 case. The VC has extremely little impact on vehicle performance. The only time it matters is when you're stuck in mud, and even then it doesn't perform nearly as well as the fully locked position of a part time case.
A 242 performs better in nearly every way, the only downside is that it requires owner input. The 129 was meant to work well, not great, but to do so automatically without any input from the driver.
I also tried rebuilding a NP 129. I was disgusted to see how poorly the VC is designed. The vast majority of 129's on the road do not have working couplers and the owners just think it is doing something. They fixed that with the NP249, but at the expense of reliability. In a 249 the Coupler must work or the vehicle has to be towed. The 119/129/229 coupler is much less robust than the 249 coupler but it doesn't matter because it forms a donut around an open differential. It's more decorational than functional. It will fail and isn't worth repairing.