Moving servers and change URL

I have never done *exactly* what you want to accomplish in moving everything at once, but I think the first of these will do that:
https://wordpress.org/support/topic/how-do-i-change-domain-installtion-from-without-www-to-with-www-in-wordpress?replies=4#post-6275183

First get your database in place at your new server with an assigned MySQL User and Password, then be sure your wp-content folder is in place, then re-point your domain to the new server, then put wp-config.php in place with the appropriate two lines enabled and with its database name, table_prefix, MySQL User and Password matching your database. The next hit at your site will set things off and it will grind for a moment or two while the database is updated, then either disable or remove the two lines in wp-config.php and go to Dashboard > Settings > General and save what you should see there with everything working. Other than the matter of uploads I had relocated at the time of the move, that all worked perfectly for me.