I don't think your 30AMP converter has high enough DC AMP available for the batteries to use. If you didn't change out your in-line fuse or breaker you probably only have 30AMPs of protection still in place between the converter and the batteries.
My three 12VDC batteries demand around 54AMPs DC CURRENT when first hit with 14.4VDC in boost smart mode charge. Then they start tapering back as the three batteries start taking on charge. After about 20 minutes or so my DC CURRENT demand is around 8AMP DC CURRENT
A fully charged 12VDC battery is around 12.5-6VDC which is what you are reading. You need to look at the DC voltages again coming out of the converter with shore power on. You may have blown the in-line fuse or tripped an in-line circuit breaker between the converter and the battery switch...
With my three 12VDC batteries in parallel I use a PD9260C 60AMP Converter...
With your two 6VDC batteries in series they can looked at as one large 12VDC battery.
Just something to check before saying your converter has bit the bullet.
I don't know where you are reading the 12.6VDC from but sounds like you are reading a fully charged 12VDC battery and the converter unit which should always put out 13.6VDC when on shore power except when it is in smart mode charging seems to be OFF LINE or failed.
My batteries are so important to me when camping off the power grid I have installed a three panel BATTERY Monitor Panel which has two 12VDC VOLTMETERS (one for each of my battery banks) and one in/out 0-75AMP DC current meter. These are direct connected at the battery bank and this is where I monitor my battery's condition

Just my take on the problem
Roy Ken