Jan Dembowski on "Moving a website with no access to myphpadmin & mySQL on new host"

Try getting a copy of the files and database via a backup plugin.

http://wordpress.org/plugins/search.php?q=backup

I've used this one and the support is pretty good IMHO.

http://wordpress.org/plugins/updraftplus/

Using a plugin that way should provide you with both a copy of the files and database without needing native mysql CLI access of phpmyadmin.