Hi scritser and welcome to iRV2. We too are Floridians, up near Ocala.
The vibration you feel could come from a couple of sources. One is an out-of-balance drive shaft and another likely one is tire balance and/or alignment. You could also have a tire that has incipient tread separation. That typically shows up as a vibration that increases with speed.
The house charger may or may not charge the chassis battery - some designs do and others do not. It takes some additional electrical components to isolate the two systems and properly control the charging, and some manufacturers don't provide them. If yours is not designed that way, you can add a system to do so. Xantrex makes one called the Echo Charger
Xantrex Technology Inc. - Recreational Vehicles - echo~charge - Product Information
Another such product is Trik-L-Charge
Ultra TRIK-L-START Starting Battery Charger/Maintainer
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL