Saints won't be eliminated if that happen.
Yes they will be eliminated.
If the Panthers and Falcons win, the Saints are eliminated and it sets up a winner take all game next week in Atlanta.
If the Falcons win Sun and then the following week, they are 7-9.
If the Panthers win Sun and then the following week, they are 7-8-1.
If the Saints lose Sun and then win against TB, they are 7-9....they lose to both Panthers and Falcons on record. One more loss than Panthers and tied with Falcons (but Falcons would have won both head to head).