Sounds like you already know what you need to do.
You are correct that you probably won't be able to access (not easily at least) the wordpress admin panel from your old site. You could try a direct IP address or some hosts offer an alternative URL, which may work, but I wouldn't bank it.
So your suggestion of finding the newest posts via phpMyAdmin is your best, and probably easiest, bet. I'd just take the HTML from the
post_content field of your
wp_posts table and then input that into the new WP install via the WP Admin interface (not phpMyAdmin), unless your confident enough to swap out the whole table on the new server with the same table from the old server. I really wouldn't recommend that though, and would definitely back up the new host's database before doing anything that you couldn't fully restore.
The other trick though is to know what metadata was associated with the post. To peruse through the metadata for that table go to the
wp_postmeta table and look for all entries with post_ids that match the post_id from
wp_post field. There will probably be more than one for each entry.
Were there images attached?
You might do a google search for something along the lines of "wordpress migrating posts between databases" or something to that effect, if you still have questions.