Most banks offer online banking and bill pay so that shouldn't be a problem. Now you need to find a mail forwarding service in your domicile state. You should set that up soon so you can get all your address changes in before you leave.
You'll have to decide on your domicile state before people can make recommendations. I'm a FL resident and use St. Brendan's Isle for my legal address and mail forwarding.
Some services can scan all or some of your mail and forward it to you electronically. If you're constantly on the move, then yes you need to figure out where you'll be in a few days and have physical mail sent there (after checking to make sure the campground or park will accept the mail). I don't move that often so I can wait till I get someplace to have mail forwarded.