A culture change has to be initiated. It doesn't have to be on the very first day but sometime early on, the new coach has to find their "Wayne Gandy's recliner" moment. The implicit message being "What we've been doing hasn't worked. At all. Adapt or move on."
One thing that might undermine this plan a bit is if any of the "gotta extend!" veterans are really locker-room problems. Not bad guys, say -- but maybe too-comfortable guys, or guys riding out the string, or guys no longer bothered by losing. If you can't waive 'em, you can demote them to second string. Without checking, I kind of think the gotta-extend crew are good dudes that will at least stay out of the way of true leadership ... but I'm not 100% on that.