Re: Electrical issues with 1973 Johnson
Assuming your motor has electronic ignition ('73 50 hp does, but there's no 1973 55hp Johnson - some pre-72 models have battery CD ignitions) you could technically run the motor with no battery, but it would fry the rectifier, which is the component that converts AC power generated by the alternator into DC power that can be used to charge the battery and power the accessories.
There's a thread in the "Engine FAQ" section of iboats with the step-by-step rectifier test procedure - it's your number one suspect.
Why the motor is drawing current, not sure - how did you determine that? Any accessories running at the time (like maybe a tachometer)?