Hi BA,
By the sounds of what you are writing, you NEED to do alot of research first, because I get the feeling you maybe thinking getting a job in the U.S. is still easy. (I'm only guessing by the q's you are asking).
NO, people don't leave SWA for AA or UA, maybe the odd one here & there. SWA people are extremely happy, but at the moment competition is intense-over 9,000 pilots on furlough (United & American have around 4-5,000). When you consider that guys & girls getting into SWA have 737 type ratings as well, up around 5000-8000hrs plus, with many on 73's.
I recently went to a job fair in the US with 6000hrs TT and a jet command, its good, but I still need more and contacts are EVERYTHING!!!If you can't get recommendations your chances are going to be very slim. Attending the job fairs give you another way, but I've been to 2 now in Jan & Apr, plus I'll be going to the next in June. If you want it while still living in Oz (which I'm doing also), you have to go hard. If you're half hearted there are plenty more who are extremely qualified and US citizens living on the airlines doorsteps.
If you want more info PM me.