Define a site in DW, setting the local root folder to some folder on your hard drive. Enter the FTP information in the Remote site section and save the definition. When you connect to the remote site, you can use DW to GET all of the files to your local folder. Then, to upload to a new site, add a new Remote server to this site definition, containing the new FTP information. Select that site as the active FTP server and save the definition. Now connect to the new server, and PUT all of your local files.
Bada bing - bada boom. You are done.
First define your Local Site folder in Dreamweaver. DW will use this folder to store your site files.
Go to Manage Sites > New or Edit site. See screenshots.
Servers: Enter your remote server's log-in details. When complete, hit TEST to see if the connection is working. If all is well, hit SAVE.
From the Files Panel (F8), click on Remote Server to show the files that are currently there. Click the green Down Arrow to GET files from remote server to your local site folder.
Here are a cople of other options should they be of interest.
1. Run two instances of an FTP program and drag files from one to the other.
2. Simply point the DNS record of the new domain to the original server.