the hosting company (Register365) put in a new area "new.sitename.com".
I would guess they added the /new/ folder as a nested installation inside the public "root" folder of whatever name (where sitename.com is installed) and that you might also be able to get there via sitename.com/new/.
this "new.sitename.com" has a completely different WP-admin login page to the sitename.com and therefore different users.
As you likely understand, that is because it has its own database.
the host company...state that...I will need to...copy all the files in the new wordpress directory over
You could, or just upload a fresh set of everything (other than the wp-content folder) from within an unzipped /wordpress' folder, then add your own wp-content folder and wp-config.php.
export the new database and import this database using SQL.
Unless the two sites are on different servers, that should not be necessary if both are in the same hosting account. The database is at the server, and WordPress simply needs its wp-config.php info to access it. However, that new database currently only knows the URL you can presently see in the boxes at Dashboard > Settings > General, and that will have to be changed in both boxes in order for sitename.com to work after the new wp-config.php has been placed.
Then edit the PHP confiig file to ensure it points to the right database.
The wp-config.php file from the new site should work just as it is no matter where you place it.
Overall: WordPress folders and files can be placed anywhere a domain will find them, then wp-config.php needs to know how to find the database, and then the database needs to know the URL.