You can import the database in your server's shell easily. Just export the current database into a .sql file, upload it to the new server, SSH in and cd into the directory with the .sql dump of your old database and run this.
mysql -u your_mysql_user -p your_database < ./your_sql_file.sql
As Brian Fegter pointed out, you can also dump the file in from your old server. Add the old server's IP to your new servers whitelist and run the following from the old server's shell:
mysql -u your_mysql_user -p -h new_server_host database_name < /path/to/sql_file.sql
Both will prompt your for a password then run.
If you just need the content, use the WordPress exporter, which will require 100% less command line fu.