Mmm I don't work with cpanel or wordpress for some time but if my memory doesn't fail me you have some options, if you have ssh access you can backup the cpanel accounts with /usr/local/cpanel/bin/backup [cpanel_username] or /scrips/pkgacct --compress [cpanel_username] /backup/ and you can download the backups from the folder /backup, in the new cpanel server you can restore from "transfer cpanel account".

Now if you can't access to cpanel or the installation is damaged and the script don't work you can backup wordpress with a plugin, I used all-in-one wp migration, is a free plugin that you can download from the wordpress repo, it create a .wpress file with all your wordpress (files + db) and you can restore in a new wordpress installation, you only need to install wordpress and the plugin the new server to restore the backup. The plugin have some limitations because the dev sell additional addons for the plugin, you only can restore backups uploading the file in the restore option and if php have upload limits you must change it according of the size of the backups (the unlimited addon remove both limitations), the other issue is that you must run the plugin mannualy in each wordpress and if you have a lot of websites it could be a little boring.

The last option is to do a manual backup, for only wordpress you need to backup the folder /home/username/public_html for the wordpress files and pictures, for the db with the command mysqldump --add-drop-table [db_name] > /backup/[db_name].sql you can backup the db, on wp-config.php you can find the db name per each wp installation.

PS: Sorry if my english is not the best