NO FOUR WHEEL DRIVE

That isnt actually 4x4 its 4x2, when you turn on your 4x4 only one back wheel and one front wheel will ever spin at one time. Unless you have lockouts.

As far as the transfer case not shifting, it sounds like a problem you should have the dealer look at.

How do you figure that? Normally they are just plain old limited slip differentials. Meaning yes, all 4 wheels can have power at the same time, however, if one of the wheels on each axle has any significant amount less traction, or resistance, than the other, that wheel will get the power. I don't understand how or why only one wheel from each differential would spin at a time.

If he's upgraded to something like a Trac-Loc (prolly not a good idea for off-roading), or a Detroit Locker, then both wheels on each axle would be locked, essentially like a solid axle, except for making turns.

Or maybe I just don't understand how a limited slip differential works... One or the other.