That suggests to me that you have a single alternator and a simple split charge relay. The relay connects the two banks for/during charging and disconnects them when charging stops, so you may very well be correct. Such relays are usually fed from the "ignition on" key switch terminal. There are complications but often the relay joins the two banks the moment the ignition is turned on, so the engine battery can feed into the domestic battery.
Batteries fail in two ways. One, chemical changes take place (called sulphation) that gradually reduce the electricity holding capacity of the battery. When this happens, a simple voltage test may well show up as OK, but they will go flat very quickly. The blanket would have drained them fairly fast.
There are indications that imply sulphation, but I am not going through them at the moment because it will get technical.
Two, batteries are designed to have a finite number of charge-discharge cycles. When this number is reached, a or some cells in the battery will start to short out, self discharging the battery. Again there are signs that imply shorting cells but if the batteries were on a modest charge the engineer may not have seen them - especially if the batteries are sealed.
You are far from alone in all this, I think such things happen to well over 50% of new boaters who buy second hand boats. Keep working at it. You can see from this there is so much to learn.