In most circumstances the fill valve is fine. Water is leaking out of the tank from one, or more locations causing the fill valve to replenish the water in the tank. One example of this could be a damaged seal, or flapper. It could also be that the replenish tube from the fill valve is below the water line, or below the stop in the flush valve. If you are using a universal lever it may be pressing against the flush valve not allowing the valve to sit straight. Also the flush valve gasket could be bad, and causing the leak.
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