I hate to say it but it might be too early to claim victory. It is entirely possible that your converter is not charging the batteries when on shore power. The systems are all working now because the new batteries are fully charged. But if the converter is not working then the problem will come back to haunt you in the next day or so.
The best bet is to use a multimeter to check the battery voltage when not plugged in and the generator turned off. Then either plug in or start the generator and check the battery voltage again. The voltage should now be higher than in your first check.
If no change in battery voltage you probably have a bad converter/charger. If you are away from home on vacation but have access to a generator then you could go to Walmart and buy an inexpensive automotive battery charger to charge them up during the day. Once you get home you can worry about replacing the converter.
