[WhiteOut]

Can you drive through the wobble or do you have to really slow down to stop it? Which transfer case do you have? (What does the T-Case shifter say on it)

If you can drive through it, I'd definitely look at the tires being poorly worn. Maybe warped rotors if you're getting the wobble when you're braking.If the rears are in better shape than the front, try swapping them front to back and see if it improves.

 

[WhiteOut]

Like Polka said, I'd fix the swaybar links first. You can buy complete OE replacements from any parts store.

From your description, it sounds like the tires could be the root issue.

 

[WhiteOut]

Explanation of how the swaybar would cause the axle to oscillate back and forth please.

Because for your claim to be correct, you need to make the point that the swaybar locates the axle and is NEEDED. You'll also need to account for all the vehicles running no front swaybar without death wobble.