You may be able to save them, if they weren't dry for too long.
Disconnect battery cables, fill with water (do NOT add battery acid), then charge one or both with an auto battery charger. This may take several days; twice as long, if you do both at the same time, in parallel. If you are unable to charge them in a week or so (depending on size of your charger), and they do not indicate 12-13.5 volts with the charger disconnected, they are probably ruined.
Regardless, take them to AutoZone, PepBoys, O'Rielly or similar autoparts store and have them load test the batteries for FREE. If they put a load on them and the voltage drops (after they are fully charged), they are definitely bad; and it's time for new ones.
Overcharging can be avoided by using a marine battery isolator switch and manually limiting how often and how long they are charged. So, a new converter may be unnecessary. It depends on how much you want to spend. However, please note that use of the isolator switch completely disconnects your batteries in storage; and they will not drain over an extended period. We turn our batteries to OFF, and the batteries have retained a charge for over six months. I suspect that they would last more than a year, if we happened to go that long between camping trips; because Optima batteries are supposed to hold a charge that long.
See additional info at this link: