It is definitely more work than just moving it from one folder to another on the same server. It's certainly up to you if you want to go through this process or recreate the site as you've been doing - it seems that you're almost done with that, but I'll put my notes about the move process here just in case you're interested.
- Make a backup of the old site's database, just in case.
- On the old site, go into the WP dashboard and go to Administration > Settings > General and change the "WordPress address" and "Site address" to the domain of the new site.
- Make a backup of the old site's files and database (which has now been modified with the new site URL)
- FTP the files to the new server
- Import the database from the old host to the new host (the exact process here differs depending on the host)
- Edit the wp-config.php file to point to the new database location
If you still need the old site to work after you've done all this, restore the old database to the old server - this will put back the old URL settings, so that the old site will work again.
Pardon me if I've skimmed over anything. Depending on your exact situation, server environment, etc., things might be slightly different, but this is the basic process.