At the new hosting account:
- create a new database
- start phpMyAdmin
Based on the errors that were listed, I'd be willing to bet that your step #1 was not done. :)
If one has access to SSH, databases can be imported by directly accessing the mysql server. This method has no limitations on execution time or file size, and files can be in any compressed format supported by the operating system, but it does require some familiarity of command-line operations.