You need to DOWNLOAD the public_html folder to your computer.
You need to follow the tutorial
You need to firstly login to the old control panel, select your site in the list, back up your files, now once backed up use the "move account" URL from the tutorial, once you do this and if it says "Account already exists" ignore this, hit back to your account list, enter your control panel of your site, browse all the way down the control panel and hit delete site/account.
Back to empty account list, hit your name on the right hand side under customer details and hit delete profile.
You will now be logged out of your account, now enter your login details again and the new panel will appear!
Click the + icon and choose a FTP username and password, now hit set web address and add your old domain.
Now upload your backed up files.
Site will be working fine after this.