Bouymaker offers sage advice on assessing the condition of the hinge pins before moving forward with latch issues.
I will add that some worn out hinges can also be very rusty. This rust will sometimes make the hinges seem tight and just fine when they are not. If the hinge pin is heavily worn and the holes are wallered out, but backfilled with corrosion, everything will be out of position and the door won't behave as designed.
