A lock or handle that wobbles almost never means the lock has failed. It means the screws have lost their grip in the wood — the hole has stripped out bigger than the screw. Tightening just spins the screw in sawdust.
The lasting fix takes ten minutes: remove the screw, coat a few wooden matchsticks (or golf tees for bigger holes) in wood glue, tap them in flush, and let the glue set. Trim, then drive the screw back into solid wood. For a door that gets heavy use, upgrading to longer screws that bite into the frame stud makes it stronger than new.
While you’re there, check the strike plate — the metal plate on the frame. If the latch doesn’t click in cleanly, no amount of tightening will make the door feel solid. Sorting locks, latches, and plates is routine work for us: CMG Bridge, Staten Island, Monday to Saturday.