Just pressing the button once producing a second click is what I am assuming you have entered. I have had that happen on my own logitech mice after a while, and it was the mechanical button inside the mouse wearing out (there is a tiny spring which weakened, and allowed the inner switch part to bounce during the movement.
Assuming it is on a PC, check the mouse isn't programmed to doubleclick (it might be a setting). You could also try adjusting the double click rate which might help not have it detect it.
If it is the button, opening it up and replacing it (which would need soldering skills) would work. The button likely has a part number on it, and likely could find it at a place like Digikey.