Ed,
No, the alternator will keep the engine running even without a battery. Many times I've disconnected the battery on a vehicle and left it run for a long time. I know one farmer who doesn't use a battery on his tractor. Just uses jumper cables from his pickup, starts the tractor and disconnects the jumper cables. No battery on the tractor at all. (and its NOT a diesel )
The generator on the other hand has to have a battery present to produce voltage.
What you're describing on your Dakota sounds like a bad or weak alternator. The electronics on these newer vehicles use a lot of current so if the alternator was weak and not charging enough to keep up with load demands the vehicle would stall.
Brad