Rather like moving furniture to new house it is required to move the webpages and databases to a new host.

• Create new SSH key pair Dreamhost
• Enable SSH access using keys Dreamhost Hostgator
• Use rsync to upload all files avoiding changing root .htaccess
• Upload MySQL database(s) need admin and user MySQL accounts
• Verify site working without going through domain Dreamhost
• Verify PHP MySQL pages working

SSH access before public key set up may need -o PubkeyAuthentication=no option to suppress use of public keys.