Tom,
You wrote "one or both" batteries.
If they are 12Vs in parallel, go buy a lottery ticket while your luck is holding. What you just had was the precedence to a battery explosion.
If they are 6Vs in series, then either a single cell in one has shorted or the converter is over charging them.
Either case, the diagnosis would be easier if you had any kind of a meter at all. Even the cheapie from Hazard Fright will do.
When you come up for air and have your new meter in your hand, Disconnect the power to the converter. Let things sit a while. Measure the batteries - This should be something a lot like 12.6. If it is a lot higher, turn on an light for a few minutes - like 5 - Now are you down to 12.6?
Turn the light off and power up the converter. Measure again.
If the measurement does not increase by at least 0.3, the converter is dead, but I doubt that.
If the measurement is ~13.3, the converter can probably be left alone.
If the measurement is over 14.4, let it be for about an hour and try again. By then, it should have dropped to the mid 13s.
If it above 14 for more than a could of hours, unplug the converter as it will trash your new batteries too.
This is all a simplification of the check I would do before the depression put us out of business. If you find something that does not fit the cases above, come back here with what you do find. Getting a manufacturer and model number for the converter will also get you more accurate responses.
Good Luck
Matt
Matt & Mary Colie
A sailor, his bride and their black dogs (one dear dog is waiting for us at the bridge) going to see some dry places that have Geocaches in a coach made the year we married.