[prairieschooner]

It most likely isn't your Batteries but you should start there, sounds like your Batteries were Charged at less than 75%. When troubleshooting I always start at one end, then the other end, then cut the circuit in half, then in half again, etc.
12.4 is very low Voltage for a 12 Volts DC System. If you have Wet Acid Batteries they should be Charged at 14.4 Volts DC, follow this link for more;
http://www.westmarine.com/webapp/wcs...ttery-Charging

I would start by checking all of the Battery, Battery Switch and other Connections and making sure that the Cables are tight at every connection.
Now let's look at your Charging System with the Alternator. I suggest that you isolate the Engine Batteries (shut down the House Battery Switch and Battery Merge for this test). I would use a Digital Volt Meter and test the Voltage at the Alternator and then the Batteries. You should see 14.4 Volts DC when the Alternator is Fully Charging.
Not being sure about the Shore Charging System it is hard to advise. I like being able to use the Charging Function of our Heart Inverter (bought by Xantrex). It has a 100 Amp DC Output making getting to about 90% fairly fast, about an hour or less when we Dry-Camp, rule of thumb- use 50% of the Battery Capacity. Now if you have 240 Amp Hour Battery Bank you should only use about 120 Amps. If you have a 20 Amp Battery Charger it will take at least 6 hours to get the Batteries up to 90% Capacity.

*The Charging Systems for our House and Engine Batteries are completely Separate, and this is by my design. My Battery Merge Switch only allows the Engine System to Charge the House System if I Manually Select it to do that. I personally like to have my Charging Systems Separate but it isn't for everyone.