When we went full time in our RV, we used St. Brendan’s Isle in Green Cove Springs, FL.
Mail Forwarding Services at St Brendan's Isle
They did exactly as you have described, scanned mail and sent us what we required. They were great to work with, they primarily work with RVers, military, traveling nurses, etc. who need an address but can’t access it physically. They can help you establish residency in FL to save you on income taxes and to allow you to vote.
I am sure you already know this, but it will be easier to set up all your bills to come by e-mail or automatic pay from your bank. It will significantly decrease the amount of mail you receive or need to keep track of. The mail service will charge you to scan and/or forward physical mail. You can instruct them not to forward catalogs and junk mail.
It worked great for us for he two years we were on the road.