Cut both sides for the new CA hinge but cut an extra deep slot in one side. Mark the center of the hinge with a pencil mark. Feed the hinge into the deeply cut side. Carefully walk the hinge over into the other side using any pointed sharp instruments that appeal to you. When the center pencil mark appears in the center it's is time to apply new CA.

I've had Great Planes hinges fail twice as you describe and this last time I used some Radio South hinge material. No report on them as yet.