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It sounds like either you have a bad 'seating' of the flapper or the tank's water is running out the overflow stack before the ball has a chance to shut off.
In the former, clean the area on and around the flapper (as well as the flapper) to insure a clean seal. In the latter, adjust your float DOWNWARDS. It's also possible that your overflow was 'cut' (yes, I've seen that before), therefore, the new refill system isn't 'ready' for that eventuality. (Either replace the pipe or, again, adjust your float downwards.)
What I'd consider doing, if I were you, is measure the HEIGHT of the overflow stack. When you visit the Home Center, see if a replacement stack on the shelf is the same height. If not, go from there. DO remember this - To avoid leaks and such, spend the extra $$$ even if you don't have to on replacement parts for the ENTIRE tank. You see, old parts don't work w/new ones, especially when it comes to water-related parts. Oh, make sure your 'chain' isn't too tight/taunt. If it's not loose enough, it can actually hold the flapper up and off of the drain hole.