I highly recommend you call Escapees headquarters to get more definitive answers to your questions. They are more familiar with RVers in situations like yours, and can also connect you to an attorney who deals with domicile/residency/small business concerns related to RVing. 936-327-8873
Sidenote- yes, you typically can use your Rainbow Drive address as your business address, too, but you need to let them know when you set up your account. There are some procedures they handle differently to help accommodate that particular need.

It may not be applicable to your business situation, but like Carlos said, having  an EIN and an "official" business (via an LLC or other means) does make a BIG difference in your relationship with other businesses that you may want to have a relationship with. Many will likely not give you vendor accounts, or give you wholesale pricing without it. For a pure tax business that may not matter, though.