Split your thinking out to 2 separate areas of interest.
No satellite TV antenna/dome with get you internet service. There is a possibility that a satellite internet antenna might be capable of receiving TV service but this is an expensive and somewhat clumsy option.
For internet service you will be best served by one of the major cellular provider. I dumped Verzion due to high cost but they do have the best coverage. T-Mobile (especially if you are over 55) has great pricing and excellent coverage.
For satellite TV your King Dome should work and will give full HD if you switch to DISH and it's working properly. If you want to stay with DirecTV (my preference) and you want HD then you either need a Trav'ler ($$$) mounted on the roof or a home style antenna mounted on a tripod beside the coach which you manually point at each new location (pretty cheap).