Why wouldn’t you just take a home directory backup from your cpanel backup area. This will create a .tar.gz file that includes all your files, email, database entries..
ftp the file to your new account at the new host and ask support to install it.
I do this all the time and it works the best for wordpress and other non wordpress sites as well.
The only catch here is make sure your user id in the new site is the same as the old userid ( the one stored in the backup )
Please, let me know if you have any questions.