The post that holds the inside handle is spring loaded and is supposed to snap back when the handle is pulled. More than likely, the spring is either broken or the tab that keeps tension on the spring is broken. Either way, the inside door panel has to come off to correct the problem.
The issue of the lock only engaging part way is likely due to the door handle not seating where it should. Again, the door panel has to come off the ensure that everything is lined up and working as intended.
I've found, too, that the latch mechanisms on the doors wear and eventually won't let the lock work as it should. Remember, the car is 50+ years old and when built, no one ever figured that they'd still be in use all these years later! You might end up needing to replace the latch mechanism...which is an easy project. Four or five screws and you're done!