Mary - Alot of us leave our trailers plugged in 24/7 when parked at the house. I have two trailers here at my CAMP BACKYARD that are plugged into my Garage 120VAC circuits. One is plugged into a 15AMP circuit and the other is plugged into a 20AMP circuit with no ill effects. We use both trailers here at the house on a regular basis as a getaway second bedroom, use the extra fridge, etc. I'm always working with my two way radio setups in my off-road POPUP trailer.
Your 2012 trailer should have the smart-mode technology converter/charger unit installed. You might pass along the model number just to be sure. If you are using this modern smart-mode technology type of converter/charger unit it should take care of your batteries automatically when parked at home. I would still set up a monthly schedule of checking your battery fluid levels just to be sure they do not drop below the cores of the batteries. My batteries are all over 5-years old now and doing just fine sitting for long periods of time here connected up to shore power from my garage. My converter/charger unit is a PD9200 series which replaced a WFCO WF8900 series converter/charger unit which would not go into smart-mode charging when you thought it should have. Perhaps you may want to change out the converter/charger unit if it happens to be a WFCO WF8900 series type unit. Pass along the manf and model number and the guys on here can tell you what you have.
You must have had a battery short out on you possibly from running without the proper battery fluid levels. I am surprised to hear that battery cables were melted down however as most battery installations have catastrophic fuse or manual circuit breakers installed right at the battery terminals to protect the battery wiring cables. Melting the cables like that can only be done with a large short circuit. Perhaps this was caused by loose terminal connections over time... I almost suspect something else has shorted the battery cables...
At any rate you should replace both batteries as mixing old and new batteries does not work out as the older battery will draw down to the new battery to its somewhat aged operating characteristics. You new battery's extended life will be shortened.
The wiring on the melted terminals and all will most likely have to be replaced as well... Just make sure all connections are solid and clean. Also make sure you re-connect your battery terminals back up in the proper positions. You hear alot of comments about how easy it is to get the battery terminals connected in REVERSE POLARITY since the same size battery terminal lugs are used on both sides of the batteries unlike how a car battery is installed. Operating batteries in REVERSE POLARITY blows fuses in your Power Distribution Panel.
Alot of folks will take photos of their battery setup connections using their cell phone cameras prior to disconnecting them so they can reconnect the cables in the proper order. I have all my cables that are connected to the POSITIVE battery terminals marked with a large DAB of RED fingernail polish...
Keep in mind the batteries is your trailer's lifeline and demands you check on them on a scheduled basis. Hope your upcoming trip has not been put in jeopardy because of this... I always carry an inexpensive multimeter in my RV TOOL box. A charged up battery without shore power should read 12.6-7VDC. Then go turn ON the shore power and the reading at the battery terminals should jump up to 13.6VDC indicating that battery connection to the on-board converter/charger is OK... These $10-$20 multimeters are available from AMAZON-LOWES-WALMART-any number of auto parts stores... Looks similar to this
Just my comments