Moved Host, Got Error

So, I’m in the midst of changing hosts, I have copied all the files over from my old server into my new one. Both the new and old hosts use cPanel and I have also copied over my SQL database. Now, I get this:

Can’t select database

We were able to connect to the database server (which means your username and password is okay) but not able to select the MY_DATABASE database.

    * Are you sure it exists?
    * Does the user MY_USERNAME have permission to use the MY_DATABASE database?
    * On some systems the name of your database is prefixed with your username, so it would be like username_wordpress. Could that be the problem?

I have a feeling that I incorrectly copied over the SQL database. The new database has a different name (I made it, then ran the text of the SQL in myPHPAdmin). Any tips?

Edit:
Oh yeah, the new host has a newer version of cPanel I believe. And I can’t just import the SQL backup, as my current host already has some other databases as well.

Edit2:
Also, on the old host my wordpress was located at MYSITE.com
but on the newsite its at MYSITE.com/DIRECTORY do I have to do through the SQL database and make the changes?