When moving a website from one server to another one and when you create links between different DB, this mean that you have inside the SQL the "foreign" DB name that is written in the MySQL statements.

So you will have to change all the DB name in the SQL file that will be used for the import.
You will have to change all the links to reflect the new DB name.
JMS is able to rebuild all the table except the "Joomla user" one that are only created when you build the DB.

The idea can be to use PHPMyAdmin to export the DB (its SQL), search and replace the "old dbname" with "new dbname", and finally import the result on the new environment.
Of course you will have to check that all links (views) are correct.

I see... well, I may still buy an hour of your time to over see this process since it's important... that is if you can.