It is not hard to change the fill valve. It does sound like that is your problem, but there is a large tube, usually right in the middle, about an inch wide, that is an overflow tube. Usually the height of that tube is lower than the handle, for just the reason you are seeing.
If you could give us a picture of the insides of the tank, we would be glad to give further advice. Right now you have a problem that could cause some serious water spillage in your house, so don't delay on this.