When I have moved a site, I use the excellent "WP Clone" plugin to backup and restore the site and content. It takes care of adjusting the URLs in the database for the new site.
I first create the new site with a generic install, then install the WP Clone plugin. Then I go to the 'old' (current) site and install the WP Clone plugin. Then I run the WP Clone plugin process, which creates a zip file of everything.
Then back to the new site, and use WP Clone to restore the zip I created at the old site. That will require a re-login of the new site. But you should be able to access the new site, and all content will be visible.
Note that the restore to the new site will overwrite all settings, plugins, themes, data, etc. Which is why I create a new site with a basic installation plus the WP Clone plugin.