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Agree with ross kernez, You'll have to build local links if you want to rank locally in google, You must update your new address and physical location, and phone number. I am affiliated with Eckerts Moving business, Along with taking care of all their moving task we always build high-quality local links for clients which are moving their business.

So about updating links to the new one. I see that you said that I should use the https://www.domain.com at all times. Then you said that the www is considered a new entity, but then you mention that it was ok to not use the www in my domain. Are you talking about the backlinking domains I do from here or on the actual website itself? Last thing.  is the backlinking rule that it should always match the exact way it shows in the URL bar? So for example https://www.domain.com should always be https://www.domain.com and never https://domain.com or even domain.com?