Copy wp-content and the rest of the files. Export the database. Import the database into the new MySQL. Point wp-config at the new credentials. Search-replace the old URL if the domain is changing. Update DNS last.
Plugins can help with the dump and search-replace. They cannot skip the database. Deploy history on WPHost restores files from previous uploads; it is not a substitute for a database backup.
Test on the preview URL before you change DNS. That is the entire reason preview URLs exist.