Much confusion here. Broncos never had a Dana 300, they only existed in Jeeps and IH, and they were always for a right side drop front axle. Broncos always had a left side drop transfer case. So the front driveshaft would never work.
Guessing you just completely butchered the model of transfer case and you really mean a Dana 20 that Broncos came with. Well that is still confusing as the shift pattern you list isn't correct for your '68 Bronco. It should be
4LO
(2LO) hidden, just a notch between gears
N
2HI
4HI
OK, maybe a J-shift transfer case? Do you have a swapped in Automatic? That might make sense.
Shifting into and especially out of 4WD isn't difficult. You just have to think about what is happening. You are sliding gear in and out of position. To get into 4WD it is a matter of the gears aligning in order to engage. If the teeth hit each other, they won't go in. You need a tooth to align with a root and they go in. Often there is enough play in the driveline that if you are close the teeth will knock each other around and go in.
Getting out of 4WD is a bit trickier. The front and rear axle likely have traveled slightly different paths. this puts the axles and driveshafts in a bind. The front has pressure turning one direction and the rear is fighting is in the other. Those gears that were engaged are now under a lot of pressure. Its all between the axles, nothing from the transmission. the gears need to be teased into not being bound up. If driving in a straight line a little blip of the throttle (on and off the power) is often just what is needed to momentarly relieve the pressure on the gears. Light pressure on the shifter during the blip will get it out of 4WD. If sitting still, change in direction will also do it. Sometimes with an automatic just putting in reverse is enough without moving.
Now that is just to get from 4WD to 2WD, changing from high to low range is a different game. Still sliding gears, but it is a crash box. It does not accept shifting while moving. Manual transmission is pretty easy, stopped, transmission in neutral, (clutch in for the bonus), shift transfer case.
Automatics are trickier. there is a small clutch drag that will spin the output shaft up to engine speed in neutral. Park works great but you cannot allow the truck to roll up against the parking pawl. That gets you back into the loaded gear issue. Stop, hold brakes, shift to park (do not release brakes), shift transfer case.
If you shift the transfer case to neutral and the transmission is not in park. You have to turn the engine off. Putting the transmission in park at this point would be like putting it in park while driving at 30 MPH. That is about how fast the output shaft will be spinning as the park pawl goes bouncing off it.
So lets start by defining what transfer case you really have.