Daniel Doezema offers a script that helps migrating the website to a new domain: http://dan.doezema.com/2010/04/wordpress-domain-change/
I’ve added MU support to that script, which is available for download here: http://dl.dropbox.com/u/1205235/wp-change-domain.php
For MU, after you run the script you must also update your wp-config.php and change the constant DOMAIN_CURRENT_SITE to the new domain.
NOTICE: My MU version wasn’t tested a lot, so BACKUP ALL YOU DATA before running it.
If your login check DOES work, I might have your solution.
When I activated the plugin, it didn’t create the database tables (didn’t run the install function). I don’t know why this happened.
Try going to http://example.com/wp-admin/options-general.php?page=twitter-tools.php&activate=true (example.com being your WordPress installation URL). This should run the activation script and create the table wp_ak_twitter. After that you can click “Update Tweets” and you should see your tweets.