Not while full time RVing, but while working and residing outside my domicile.
The greater issue was state income taxes. My employer set me up to be paid by a foreign corporation while working in a third country. Federal IRS could handle this, filed Federal using foreign tax credits (my income tax rate was as high as 80% where I worked). State of domicile did not accept my non-residence, wanted taxes as if I was living in my domicile. So make sure you understand tax laws for your chosen domicile and any place you might work or get unearned income.
Second issue was licensing. To get a driver's license where I was living, I had to surrender my license issued in the U.S. My U.S. license would have been usable as a visitor, but not as a resident with work permit, visa, resident permit et al. Fortunately my U.S. license was held escrow, so I could retrieve it for trips to the U.S. if I didn't want to try driving on a foreign license as a visitor.
While this was having to do with residence outside the country, both issues can come up state to state, domicile vs residence, if you are not careful and a state you are visiting decides that you are a resident. The tax and licensing things come up particularly if you are working, but being someplace too long not working can also get attention.
I ran into them during six years in the military, Viet Nam era, because I kept my home of record as my domicile. I had the protection of what was the called the "Soldiers and Sailors Relief Act" but even with that I had to have the JAG office go to bat for me in one state over the driver's license thing. Another state got touchy about my using a military drivers license in an unmarked government vehicle off base, but I was able back it up with badge and law enforcement credentials.
Your problem will not be with your domicile state, rather avoiding becoming a resident of someplace you are visiting.
With respect to the bank questions, I've started making a break with local banks and credit unions that have been handling my money the past 35 years, by setting up parallel accounts at USAA (where I've been doing business for 48 years). It can be important to close out accounts in places you no longer wish to associate as eiter residence or domicile, and it helps to have established a prior relationship with a bank that operates globally when you want to finally make the break with your past.
