I have no clue how Dreamhost or Fatcow work, so I can't give you any advice on getting to your hosted databases there unless they use a standard interface like cpanel to access your files. If you have cpanel access, you can get to the export and import functions with a utility they provide called PHPMyAdmin.
If you can't get to such tools supplied by your host provider you'll need to figure out how to grant outside access to the databases (again, that depends on your host and you'll need to ask them how to do it), and install your own tools, such as the one I first mentioned--actually, I just checked and they've revamped their tool set so what you'd want is the Workbench tools:
http://www.mysql.com/downloads/workbench/
Seriously, this really comes under the purview of technical support from the two host providers you're migrating between because they might have particular issues there's no way someone outside would know about. Have you contacted them for support yet?
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