I replaced the flapper, and everything was good for almost a week. Now, it sounds as if its running constantly, but I don't see any water being released or the tank constantly filling up. The toilet flushes and refills just fine. The tank refill sound is different than the constant running-type sound.

Close the stop under the tank, wait an hour or so, do you still hear running water? Could be a badflush valve rim that the flapper rests on but then you'd hear the fill valve lick on every once and a while