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I've done this before. Pull handle out as if opening the door and hold in this position while door is sprung open.
Look at the inner backside of the handle and you will see a small retaining screw.(Phillips) They usually don't loosen up.
What happens is the ears on the lock mechanism that the handle slides into fracture and the handle becomes loose and wobbles up and down.
Reason: years of pulling the handle up when opening/exiting instead of straight back towards you when opening the door
This is what wears out/breaks the pot metal ears on these castings.
I just purchased two take offs from eBay that were like nos some years ago and that fixed the problem.
Door panel needs to come of to replace the mechanism and then its straight forward from there.
Think of how many times these have been open/shut over the last 50 or years.
Marshal