Even after I replaced the flapper, my toilet continues to run (until I shut off the supply). But the tank water is not dripping into the bowl. Then why do I hear water running? Is that even possible? What should I try next?

The flapper only controls the water that is held in the tank and gets drained into the bowl when flushed. The fill valve controls the water that enters the tank and sounds like your problem.

The tank water can go into the bowl two ways. Through the bottom flapper when you flush (or if the flapper does not close completely due to deformation and age or the chain got stuck being too long), or it spills into the overflow tube when the tank water level rises too high.

If as you said the tank is not dripping into the bowl through the flapper or the overflow tube, and you hear continuous water running then your tank water will rise higher and higher until it flows into the overflow tube or spill over the tank top onto the floor.

If water is indeed flowing into the tank from the overflow tube then it is either the overflow tube is set too low and needs to be adjusted higher, or the full valve needs to be adjusted so that it stops filling before the water level reaches the top of the overflow tube, or the fill valve has gone bad and leaking spraying and not stopping.