Is the shutoff valve fully opened? If it is old and has a washer, replacing it with a new 1/4-turn ball valve may solve the problem. Replace the supply hose at the same time as a precaution.
There could be some crud caught in the valve, but that could prevent it from shutting off, so replacing the seal probably wouldn't help. It's easy to take that apart, though, and you could have something caught in there that might flush out when removing the seal.