First, try swapping batteries with the good unit. A couple times in my life a "new" battery I've purchased turned out to be pretty much depleted.

see other post

Try programming it this way. Press and hold the button on the remote while you press and immediately release the smart/learn button. If this doesn't work and your opener was manufactured between 1993 and 1997 you have a bad logic board.

Now I can give you the information that you really don't want to hear. The code is in the remote and every time you program it you are setting the same code in the receiver. The only way to change the code is to change the remotes. The odds of someone else operating your opener are so high it is just short of impossible. I would look at the wall control and the attached wiring. The wall control can go bad over a period of time and cause problems like you are having. So can the wiring. Sometimes the staples used by the installer can wear through the insulation and cause intermittent shorts that act just like pressing the button. Hope this helps you.