Something is definitely wrong, as the windows are only supposed to operate when the key is in the "run" position.

Have you tried it with the fuse for the windows removed? If they still work with the fuse out, a previous owner (or their mechanic) has moved the main power wire for the windows to a different fuse---one that is hot all the time. Perhaps the fusebox had to be replaced---melting of the #16 fuse socket is not uncommon---and the wires were replaced incorrectly. If this is the case, remove the other fuses one by one to determine which one is powering the windows. Note that it is a somewhat involved task to get to the back of the fuse box to swap wires around. The entire lower dash has to come apart for access.