It is recommended that you use the program
PhpMyAdmin, often available from your web control panel. From here you can Export all of your ListMail database tables, and data, into a zipped text file file.
On the new host, set up a new database and, again through PhpMyAdmin, import / upload the data file to set up the tables and data in the new database.
Next, upload the ListMail files to your desired directory on the new host, ie
public_html/mail, and modify config.php to contain the new server's MySQL access info.
You should then be able to access ListMail on the new host with all of your old data. If necessary, update the path to ListMail, the domain name, and re-setup the cron task and bounce scripts.
http://listmailpro.com/help