UPDATE: Doesn't seem to be the converter. I took the advice of several on this forum and didn't even ask to test the old battery. I just went ahead and purchased two new 6 volt batteries (Trojan brand) and plastic housing for each. installed the batteries and everything seems to be working good. We've run the heater several times and it's no longer cycling the fan speed from low to high. Thank you for all the help and advice. It's greatly appreciated!
P.S. All this took place in Idaho Falls, Idaho. We are now in Bozeman, Montana on our way to Minnesota. :)