ABS light with buzzing under the hood

94 Maxima w 180k. ABS light came on and then a buzzing under the hood while driving and then buzzing continued after ignition was turned off. After about 15 min I disconnected the battery and the buzzing stopped. I reconnected the battery and the buzzing had stopped. Research on the net brought up that it was the ABS relay. The car started the next morning with a 20min drive to work and then it sat for 16 hours. I tried to start it but the battery was dead.

The battery is not new so I doubt it is taking a charge. Could the relay be draining the battery? Is that the problem or could it be something else?

Thanks Andre