Based on the volts gained with the engine running the charger has failed and can no longer produce enough DC power to meet the demands of both the coach systems and keeping an adequate battery charge. Probably the original that was installed when the coach was built. Replace it with a modern multi stage charger. I'm a bit partial to
Progressive Dynamic PD9200 Series chargers. And considering the coach is a recent acquisition having the 6VDC batteries checked out at the local auto parts shop wouldn't hurt.
Regarding what to replace the batteries if necessary? Depends a bit on how the OP plans to use the coach. Lots of back country off the grid away from it out camping, the biggest, as many as you can fit and afford 6VDC will be a good start. Every time out staying at campgrounds with full services then a single marine 12VDC will work. Split the difference and stay with two golf cart 6VDC.