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If the motor is a 12V motor, a bad battery would make it shut off. If it is a 24 or 36V, one bad battery would only make it slower.
since it is just dying, you will need to have the head off of the motor while you run it, when it stops, test the voltage on the red and black wires while having the motor on Const, high speed, if you have full voltage there, it is the motor that is bad. If you do not, it could be the remote or the power supply in the boat.