The battery disconnect switch, also called the Use/Store switch, on the wall controls a solenoïd which does the actual connect/disconnect of the batteries to the
12V circuit. When the batteries are connected to the
12V circuit the converter which feeds that circuit by converting shore power to 12V, charges the batteries.
The solenoïd switch magnetically moves a metal bar in one direction or the reverse direction between two poles to make or break the 12V circuit connection with the batteries.
In order to activate the magnet in one or the other direction, there are two 5Amp fuses on either side of the solenoïd, one to move the metal bar in one direction and one for the other direction depending if you press the toggle switch on the wall to Use or Store.
The first step is to test those fuses to make sure they are Ok if not the solenoïd won't switch, which may mean the batteries stay connected to the 12V circuit, or they stay disconnected from the circuit, depending on which fuse is cooked and the position of the bar when it happened.
The filament in a 5V fuse is quite fine and eyeballing may not be sufficient, it is better use a multimeter to check it (a very useful tool in an Airstream).