Can’t login need to move site

I see it appears likely I should be able to do this but I wanted to be sure of what I’m thinking. I have allowed my domain name to expire and so I registered a ned domain name and I’m hosting it on the same server I had the original site BUT I can’t log into the original site I can only access it by way of FTP. So I don’t see how to export the various things like the database or even the website itself except to copy & paste the file structure from its original folder to its new location i.e. New (live) URL same server. Hostgator setup.