Helping Moving MySQL Databases

I am in the process of switching hosting companies and I was wondering how to go about transferring two MySQL databases, each of which are about 1.3mb in size. I am somewhat familiar with PHPmyAdmin and I’ve tried exporting the database; however, when I try to import on the other server it’s giving an error because it’s trying to create a brand new database.

Similiarly, another problem involves the naming of those databases. My current host has this named like this:

name_blog

And my new host has them named like this:

name1_blog

I want to make sure that that change, while minor, will not affect database or table structure/content. Any advice you all could give would be much appreciated, I’ve tried doing Google searches but haven’t had success thus far.

 

Can that be run inside phpmyadmin or is that from the command line <-- my current server doesn't allow command line access from what I understand

That's a good idea, I didn't think about just exporting the tables, not the entire database.