Hi,
If you want to move your AcyMailing data from one website to another, you should:
1/ Install the latest version of AcyMailing on your old website (that will make the migration easier)
2/ Copy the folder media/com_acym from your old website to your new website
3/ Move all the #acym* 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 to the one you use on your new website.
So once exported, you should open the file and replace the text "#_acym" by your new prefix (for example bjky_acymailing) and then import the file on your new database.
4/ Install AcyMailing on your new website
5/ Login to your account on Acyba's website and change the domain that is assigned to your license.