the live well fill/empty/recir. are operated mechanically by your switch via a cable to a valve. the valve is usually located outside of the livewell between live well and the transom , it could be a broke cable or broken plastic arm on the valve, but most of the time it is just a clog in the drain hose. what you can do is put your switch on drain find the outlet for each live well (usually under the motor on the transom) put a water hose to it and try to back flush the line. if that doesn't work your prob. gonna have to check the valves to see if they are working. google flowrite live well valve this will give you an Idea of what they look like.
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