Banking and credit cards can stay the same. However, you need to give them your new address. That is done by establishing a mail forwarding service and it's recommended to do it in one of the popular full-timing states such as Texas, Florida, South Dakota. You will need an actual street address not a P.O. Box. That doesn't mean you have to live there or even stay in those states. It's just an mailing address.
We've belonged to the Escapees RV Club mail forwarding service for 17+ years and would highly recommend them. They give you a street address that has been proven to be very legal via the courts.
It's a very easy process to request the mail that the forwarding service is holding for you. In our case if we know where we're traveling to we find out the nearest very small town post office address, zip. Then we ask the forwarding company to send our mail c/o General Delivery to that post office. It is then sent and we can count on receiving it within a 3-day perios via Priority Mail upon our arrival. We've never had a problem doing it this way - even on our trips to Alaska.
We stay in public parks (national, state, forest service, etc.) so we're not having it sent to an RV park but instead, General Delivery. Also, be sure to find a very small town for the General Delivery post office. Some bigger cities have multiple post offices and is not a reliable way of getting your mail.
We've never had any issues in 17 years. We pay $85/year for the service and then our postage. We can have it sent daily or whenever we request it. We usually have mail sent every 3-5 weeks - we don't get much nowdays as our bill-paying is done via automatic deductions. Newly-issued credit cards and some financial/medial documents are sent to us this way.
Have a great time in your new lifestyle!