Battery dead when converter is on

[Phoenix]

I have a 2005, 19' Bambi, and the two 12-volt batteries go dead after a couple of weeks with the converter running.

We camp one or two weekends a month, so we leave the trailer connected to 110-volts AC all the time when it is parked at home. You'd think the batteries would always be ready to go, but after a couple of weeks with the converter on, they are both dead. This has happened several times. By the way, the batteries did not boil dry from being overcharged.

Is there something wrong with the converter, or is it likely that I need new batteries, already? All fuses and connections are secure, the batteries show 13-14 volts at the terminals, all cells are full of water (acid) and when you disconnect the AC, the batteries are dead.

The trailer was purchased new about 18 months ago, and the batteries were manufactured a couple of months before that.

I seem to recall that the converter isn't the best way to charge the 12-volt batteries. I have an auto battery charger. Is it OK to hook this up at the same time the AC is connected to the converter? Or, is there some other method that works better than using the converter to charge the batteries?

[ALANSD]

I would suggest a disconnect for the batteries when you are connected and they are already fully charged. Then you flip the disconnect and use the batteries as needed. Most times you are connected anyway and won't need them.