What should I do to renew my Marriage Visa after moving to Phuket without a local bank branch?

This is what you need for documentation:

The full-service office (including Extend Marriage-Thai Spouse) is the Phuket Town Immigration Office, Room 103, enter from outside on the ground floor just past the drive-through window. There are fixed chairs outside for waiting and usually an immigration officer is stationed outside to check you have all the necessary paperwork.

The Patong Immigration Office is mostly for servicing short-term visitors (tourists) and while they do 90-day reports and certificates of residence, they do not do long-term extensions and in general they do not like to see holders of long-term extensions. I highly recommend the Phuket Town Immigration office, which in this case is the only one you can use. There is also a satellite office in Blue Tree for the north of the island but it is also only able to handle short-term visitors not someone looking for a long-term extension.