If I were Matt I would pull an Eli/Elway move most definitely. Jim Haslett walked into Tom Benson's office and asked him to either A) Show Haslett that he still has a commitment to winning, or B) Jim would ask to be fired.
Jim ended up asking to be fired, and his wish was granted. Safe to say, Jim got no sign whatsoever that Benson was committed to winning.
That franchise is and always has been a mess from top to bottom. You should see some of the stuff that Kyle Turley wrote about the franchise back when he moved from New Orleans to St. Louis. He talked about how they use to try and serve moldy food to the players and stuff back in the day and everything.
Its awful over there, and now it is even worse because they are a homeless franchise, and Benson and Tagliabue are in the middle of a war determining what the future of the franchise is, where they're going to relocate, etc. If I were Matt Leinart there is NO WAY IN HECK that I would stand to be drafted by the Saints.
BUT, that said here is how I would handle it. PRIVATELY. I would tell the Saints behind closed doors that listen, I'm not convinced you guys are even committed to playing winning football. I don't know what freaking city you're going to be in when we go to camp, I don't like the idea all that much of playing my home games in two different cities or even worse, playing them at the opponents stadium and CALLING it a home game, and I am convinced that there's nothing I could do to lift this franchise from the depths. I will not play for you. You can draft me, and I will not sign. We will wait until the next draft. I already snubbed the draft once in favor of a third national title in college, ballroom dancing, and partying with the many fine young females here in southern california who treat me like Colin Ferrell. You think I won't forego the NFL again, to make sure I don't play for a franchise in the midst of as much turmoil as you guys are? I would, and I'd account myself lucky for the right to do it. But, here's what I'm going to do for you. I'll play ball, and I won't say anything publicly. You can feign interest in me, and I'll pretend that I'm willing to play for you, that way you can legitimately get a team (like Tennessee) to trade you a load of picks to get to your spot in the draft. That way you get great value in trade, and I get #2 pick money. Everyone wins.
Forget all the righteous crap about what's right and what's a "d!ck move" or whatever. This is a business and I'd make a BUSINESS decision. Teams make those BUSINESS decisions all the time without much heed for the player's feelings etc. All sides in the industry are blessed with certain kinds and amounts of leverage to get what they want and nobody can blame anybody for using that leverage.