If there isn't water flowing into the toilet bowl somewhere then it is flowing onto the floor.
There are really only 2 choices. Your float is set so that it is above the overflow or your flapper is not holding. If you have a ball-type float, bend the arm slightly so that it will turn off at a lower level. If you have the kind that slides up and down on a column, you can adjust the float down so that it pushes the arm up higher. Otherwise, replace the flapper 5 or 6 times until it stops.
There is a fill tube that you can buy for under $10 that will ignore the flapper and only fill when you flush. It often means that you have to flush, wait for it to fill, and then flush again while you watch the kids in the pool.
I will slap the **** out of Mileslong for that sig. God damn that is horrifying!
But a new ball and cock assemply at Home Depot for about $20.00. Install it and should fix the problem. Your flapper could be ****** or could be that the whole assemply is worn out like Mileslongs sig is. Just replace the whole damn thing and be done with it.
Oh, be sure you dont do what a frined of mine did when I told him to do this, and leave the water turned on. He flooded his house, and his old lady called me yelling at me for telling him he needed to do this. Of course, it fixed the problem, but created a new one.
Also, be sure to talk to those guys at Home Depot, because they can set you up with the tools you might need to do this. I got in there the first time I had to change one out, and needed a different size of pliers to get into the actual tank itself to loosen the fittings.