I think New Orleans could go a few different ways...
1. They can trade Chris Paul to the highest bidder next season to a team like Dallas, the LA Clippers, or the Knicks. It would be a fiasco again if Paul decided he wasn't signing anywhere but New York and forced his way out, but I don't see that happening. If it did, you'd be looking at a Billups/Fields/1st round pick for Paul swap or something of that nature.
2. They can dare him to walk away from the team and go somewhere else, trying to make a championship run with what's there. If the Hornets do well in these playoffs, Paul will probably just stay.
3. They can dump bad contracts like OK4 and Trevor Ariza in order to get enough flexibility to enter the Dwight Howard sweepstakes. In this scenario, New York could trade Billups with some combination of Balkman, Walker, Turiaf, Fields and Douglas and bring back Ariza and OK4 in exchange. This is a good option for the Knicks, especially if they can draft a point guard and/or a big man.