Manually Move WordPress From Host to Host

Hi All,

Hope you are all not too full of turkey, roast hams and wine to help me out once again.

I am moving some standalone sites from one hoster to another (HostA -> HostB).

On HostA I have SFTP, cpanel access (inc myphpadmin), no commandline access.
On HostB I have an admin interface and commandline access.

I have manually copied the WP directory structure from HostA to a local filesystem.

I have two options for my deployment (I might do either or both - not sure yet):
- Move the domain name to new host
- Use a new domain name

What are the steps I should take to migrate and get the site working on the new host ?
How can I export the mysql database ?
Do I just export all the tables and import them into a new database with the same name and user ?
If not, then what ?

Caveat - I need to do this manually, snapshot is not working (thats another story).

Thanks y'all !