I hate to be the bearer of bad news but I don't think that the Saints were screwed by this. The Saints were willful in violating the rules, got caught and are getting hammered. That's how this works. I wrote my third article for The Classical, which I hope you'll read, and, again, I'll excerpt it here:
It doesn’t help that, once the Saints were caught, they acted in ways that made it effectively impossible for the league to be lenient. The team probably could have mitigated the penalties by calling the entire bounty system an elaborate scheme to enforce the classic Al Davis Theory of Quarterbacks Going Down And Going Down Hard, but that’s not the path they chose. Instead, when the league notified the team that they had heard about the program, the Saints apologized and agreed to better supervise the players. The supervision system was a scheme of plausible deniability. Assistant Head Coach/Linebackers Coach Joe Vitt was tasked with watching Williams and letting his superiors know if he went rogue. Vitt watched, saw the bounty system in action, and said nothing. Vitt should have been the star witness, the person who could connect Payton to the bounty scheme. But according to Vitt, Payton and Loomis, Vitt never ratted out Williams internally. You can see why the NFL was offended that the Saints tried to chump them like that.
Thanks for reading this far. The article also covers the penalties themselves and why it was necessary for the good of the league for the Saints to get hammered. Again, I hope you'll read it.