I’d suggest:
- ‘freezing’ new posts/content/changes
- copying entire file structure of the sites from one host to another
- copy all site’s databases to the new host
- reset credentials on databases on new site to credentials on old site
- set your local ‘hosts’ file to redirect your domain.com to the IP address of the new place
- you might need to temporarily change htaccess on the new place to use IP address rather than existing domain name. Change it back when you get your nameservers pointing to the new host.
Because you copied the database and files from old to new (ensuring credentials match), the new site should work just like the old site. And media links won’t have to be changed, since your domain name is not changing.
There are ‘cloning’ plugins available that will work. I’ve used one called “WP Clone” mostly successfully.