Is this a new toilet? If it is the water level may just need to be adjusted. You've got the ballcock assembly in this particular model (carusoe maybe?). If I understood your post, the valve isnt shutting off and water is running down the overflow tube. There is a screw on top of the ballcock that can be turned clockwise to make the valve shut off sooner. Adjust the water level to the water level line on the overflow tube. If its an older toilet, you might need to replace the fill valve.