I had to replace my fill valve because the tank would not refill. I also replaced the flapper, because I believed I had a leak, because the water wouldn't stop running through the fill tube. It is not running fast, just trickling through, but I think it is running my water bill up. Anyway, replacing the flapper didn't work, adjusted the chain, thinking maybe it didn't have enough slack, but that didn't work. Any ideas on what the issue may be? Thanks.

Depending on wich type of float you have in your tank ball or sliding there is usually an adjustment in a screw fashion that will adjust your float so your water that runs through the tube will shut off, try adjusting that screw that operates your float so the water turns off earlier.

If I understand correctly the fill valve has shut off but you have water coming out the fill tube into the overflow pipe.

The fill tube should be attached to the top o the overflow pipe, not down in it. If the fill tube is stuck down into the overflow pipe and goes below the water level you will get a syphoning action when the fill valve shuts off.

Have you adjusted the ball cock so the float shuts off the water when it reaches a level of 1" below the overflow tube? If you did all this and there is still water running out of the fill tube you have a bum ballcock.