Sarah,  You can just change the domain name on the site if that is what you are trying to achieve.   No moving of files necessary.

Add another alias for the new domain name on the exising site settings (Host - Portals - Alias, or Admin - Site Settings - Alias).  If the new domain name is pointing to that same server, it should work for you.   When all is working under the new domain name, you can then login under the new domain name and remove the old alias from the alias record.

Unfortunately, I cannot access the current site due to some problems with the company that manages the domain. 

My host has said that I need to edit the config as it points to the old domain, but there is nothing in there that does this - is there something I need to change in the database?

Sarah