French doors are falling apart and don't close
Without seeing the doors, this is speculation, but I'd probably want to pin the joints with approximately 12" long 5/8" dowels epoxied in. This assumes that the door is solid wood, not particle core with a veneer skin.


remove doors
square and clamp
drill 2 holes per joint from the stiles with long auger bit. The adventure here is that if you stray off center, you've destroyed your door.
epoxy in dowels, trim, apply protective finish. 


This isn't rocket science, but it's high risk. You might consider new doors, as they aren't terribly expensive.