I don't think the league would realign. The Saints-Falcons-Panthers have been together a long time. And I think the league recognizes at least with the Falcons-Saints rivalry that it goes back over 30 years. Of course it's not he same as the Redskins-Cowboys or Bears-Packers, but I don't think the league would disrupt such a rivalry, especially when new ones are already forming with the latest realignment in the west.
Also think about how "weak" the NFC West would be without St. Louis. The 49ers are down and out, the Cardinals can never be relied upon by the league to be a consistent winner. So that just leaves the Saints and Seahawks, two more teams that are historical losers.
I'd rather see the Saints move to San Antonio, but the league has been working for a decade to get a team in L.A., and they'd do everything in their power to block a move to San Antonio.
Would the Saints rename themselves the Angels if they moved to L.A.?
"Vincere scis, Hannibal, victoria uti nescis" -- Maharbal, 216 B.C.E.