Yes. Your Dream has a built-in charging system that charges the house and Chassis (starter) batteries. Nearly all RVs have such a system, though in some older motorhomes the chassis batteries do not get charged.
On your American Coach both the chassis an house batteries will get charged. When plugged into shore power the house batteries have priority and they will be charged until fully charged then the chassis batteries will be charged. When the house batteries charge level drops to a certain point they will get charged. When driving the Chassis batteries become priority and they will be charged until fully charged then the house batteries get charged.