OK, here's the scoop....
First, you obviously need a new battery. Procure and install. Turn OFF all your DC loads on the boat- lights, refridgeration, etc.Then turn on your charger. You didn't say what brand or features that your charger has, but if it has an ammeter or at least some LEDs to indicate what it is doing, see what the charging current is.With a brand new fresh battery, I would expect that the charging current would be about 5-10A for no more than an hour, then it would drop to about 1-2A. If your charger has LEDs, it should show "bulk" or normal charging for the same hour and then should show an indication of "ready", "float", "complete", or something similiar that indicates that the battery is pretty much fully charged.If the charger is truly DUMB, then you need an ampmeter to place in series with one of the leads of the battery. Many hand-held multimeters have a 10A or 20A scale that can be used. This is something that you should have onboard (IMO) anyway. Measure the charging current after the system has settled out and is no longer decreasing. I would expect a reading of under 2A.Now turn off the charger. Connect an ampmeter in series with the battery if you haven't already done so.Then, turn on the refridgerator and lights. I know that you said the battery was a starting battery, but there could be a mis-wire such that the starting battery is somehow connected to your house bank. If the current does not increase from nearly zero, then the battery is not connected into your house circuits. That would be correct.Now turn off the refridgerator and lights. Turn the charger back on and you may see a small jump in the charging current for 5-10 minutes.Then turn on the refridgerator and lights with the charger turned on. HERE IS THE KEY:If the charging current on the starting battery increases when the DC house loads are turned on, then the charger that is charging the house bank is providing a "mirror" charge to your starting battery. IMO this is not real good. The reason is that when your boat is at the dock with power on and the charger is running, everytime the refridgerator kicks on, the charger will increase it's output to supply that power to the house bank AND at the same time it will increase the charging current to the starting battery. But since the starting battery doesn't need charging, this "excess" charge will serve only to boil off the battery.How to fix? I would not leave the power plugged in when you are not aboard. I know this means having a warm refridgerator, but it eliminates several possible problems. First is the overcharge of the starting battery, and second is the potential for electrolysis issues with your underwater metal. The boiled battery is the least of the issues.Another fix would be to disconnect the charger from the starting battery. For a battery in good condition and no continueous loads, there is no need for a charger to always top-off the starting battery. The engine will do this when it is run, just like your car. Then you can have your cold beer and you only have to worry about the electroylsis. <grin>Sorry this is kind of long, but I thought it might be useful to you.