Sounds like the handle rod needs adjustment. When you lightly push in the door button to where it loses free play and feels firm, the button should no need to be pushed very far in. If you have to push he button almost all the way in before it stops, it needs adjustment. Above the door latch there is a little hole in the door sheet metal. This is an access hole to the adjuster. It is an Allen key ( hex) loosen it, and adjust the rod so the button moves less before it makes contact.
And you are dealing with a door that has been opened hundreds of times over 41 years so its bound to be worn, loose, sticky. Removing it, cleaning and relubing will work wonders.