Loomis: Next Saints Head Coach will decide their own staff and direction at Quarterback

You can "easily" cut Carr this year despite the issues discussed. (Not suggesting you should, but that you could.)

The Saints are $52M+ over the cap right now.
Those 4 moves alone get you to cap compliance. But this in my opinion does as little damage as possible to the future cap. (Make no mistake, there's some damage that gets built in, but at least the extension is minimum to a guy like Taysom who, health-pending, has probably the most value of those guys if you don't value Carr.) [Edit for clarity: Note this is post-6/1 compliance for clarity. As I mentioned in a later post, you'll still have to find some savings with extensions to players. The team will have to make decisions on a guy like Olave or Shaheed or Juwan Johnson for example. They need to hit on whatever extensions they make.]

You'll need to clear room for some draft picks, etc. But there's still plenty of options depending on your preferences. (Not advocating, just giving options.)
Total of those would be about $13.5M. Obviously there are other guys to cut, but at some point you're just replacing those salaries with the same minimum salaries.

You can start to see older contracts come off the books then in the following years with no future penalty, assuming you want them to. Cam, Carr, Mathieu, D. Davis. It means going young and lean for 1-2 years, but it can be done if you want. I'd even argue the easiest path to compliance has to be to cut Carr with the 6/1 designation. Otherwise you're extending him.