I've run into this trouble before with both automotive and marine systems. Many will disagree, but here it is; your alternator, whether car or boat, CAN NOT charge a completely dead or bad battery. It will, however, work itself to death trying to. The charging system was designed to top off a serviceable battery, putting back the relatively small amount of power that was used to start the engine, if the battery is completely dead, or bad, say with shorted plates, it will quickly overload and burn or short itself out. What you describe fits this scenario perfectly.
Hope you can get going without too much expense.