This is the procedure to follow:
1/ Install the latest version of AcyMailing on your old website (that will make the migration easier)
2/ Copy the folder media/com_acymailing from your old website to your new website
3/ Move all the jos_acymailing* tables from one website to another (you should first export the database tables via phpMyAdmin and then import them on your new website)
Be careful, you may have to change your table prefix from jos (or another one) to the one you use on your new website.
So once exported, you should open the file and replace the text "jos_acymailing" by your new prefix (for example mnpo_acymailing) and then import the file on your new database.
4/ Install AcyMailing on your new website