There's only one Database needed for PHPBB, so its possible the others are either backups, or maybe for Mods we no longer use?
I don't know of any way to reliable transfer a live board. In an ideal world I would bring it down to maintenance mode in PHPBB, with a message saying that the board is temporarily suspended. Then transfer to the new host, wait until the DNS is updated and everything points to the new site, and finally take out of maintenance mode. I'm not sure you'll have that option with the current flakiness.
Whats likely to happen is that while you're bringing up the new board and waiting for DNS to update, a few people will see the old board pop up occasionally, and those posts will be lost.
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