If the safety is "falling off", it would make me think there is a plunger tube problem. Either the spring is weak, missing, or the plunger is bound up (or missing) and isn't locking into the little detent pocket in the thumb safety.
The test for a thumb safety is: With an empty gun, hammer cocked, holding the grip safety in, and thumb safety switched up; squeeze the trigger firmly. In your case, you will have to hold the thumb safety up into the safe position. The hammer should not fall. Release the trigger and grip safety. Hold the gun to your ear and pull the hammer back. If you hear a click from the sear, the safety is out of spec and should be replaced. Under no condition should the thumb safety come off when the trigger is pulled.
Remove the thumb safety by cocking the hammer and moving the safety to the sweet spot about half way between "safe" and "fire". Pull the safety straight out. Inspect the plunger tube. It should be "riveted" tight against the frame. The plungers on both ends of the spring should move freely in the tube. When the front plunger is seated, the spring should be long enough to push the rear plunger out of the tube about 3/8" (without the thumb safety in place).
To reinstall the thumb safety, cock the hammer, insert the thumb safety and find the sweet spot. Use a thin blade screwdriver inserted between the frame and thumb safety to push the plunger into the plunger tube. Push in on the safety and remove the screwdriver.
Just for grins, I pulled the plunger spring (and plungers) out of my Colt. As soon as I apply a little trigger tension, the thumb safety falls. The safety works perfect and is "in spec".