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EDIT: This post about virtual mailbox services might be irrelevant because banks will flag that they're still a business address. See next comment.
Look at setting up a virtual mailbox service. I use anytimemailbox.com and vouch for them. All their locations are run by different operators, so look at the operator and check if it's a legit business. (Some are mom and pop operations run out of a home.) There's also iPostal1.com (another contractor service--different locations available but not as great a user interface), Earth Class Mail (more expensive and forwards mail to a facility before processing which adds delay time but HIPAA certified), and plenty of other virtual mailbox operators you can find by the power of Google.
One thing you need to find, is that whatever address you end up with, it needs to verify on the USPS website. For example
1000 Main St
Unit 123
Somewhereville, AL 12345
is golden, but
12345 Main St
Unit 123 # 45678
Somewhereville, AL 12345
Will be a PITA because address fields aren't always designed for the double number on the second address line.
Also, whoever you choose, you'll need to fill out a USPS PS-1583 Form, which will require online notarization (usually $25) and two forms of ID, usually a driver's license and some other form of ID.
Whatever address you choose to go with, just be prepared that it will go on your credit report and become part of your public record. Personally, I've moved a bit, and having a fixed address for this information to be attached to has been a convenience.
Maybe you're looking for this answer. What I posted above won't get you around the place of residence requirement unless you choose a residence as your mail handler. My banks recognize the address I give them as a business address and they still ask for a residence address. It's a federal requirement for accounts. But here's the thing. You can give them your friend, sibling, parent's, whoever's home address, and then request all your mail be sent to a separate address. Maybe that would have been the shorter answer. (Unless you find a virtual mailbox run out of a home and that does the trick.)
Somehow I may have also gotten past part of the difficulty you're facing by first updating my addresses with current cards to the virtual address, getting that on my record with bureaus, updating my documents to match, and then planning to use that address with new applications. In general, P.O. boxes are first to be blacklisted, UPS boxes second. Years ago virtual mailbox providers lobied to keep from being blacklisted for something or else the other (I forget but it's important for business accounts.) YMMV but I hope it helps.
If all else fails I'd try getting a virtual mailbox address and then updating your Drivers License to match the address.