Echoing Howard’s advice - what exactly was the process? My crystal ball is a bit dusty!
Did you do this migration manually or use some automation script? Did you backup and restore the code base and database? Did you start with a new code base and just move the database? Is the database new and you’ve imported the sql?
Are the new database and DB user names the same as the old ones?
Did you edit the config file in the new server to reflect the new location?
Are the permissions for moodledata/sessions definitely set to 777?
What version of Moodle are you running? What server and OS are you running? Windows, Apache? PHP, MySQL version, etc.)
Did you upgrade to a later version of Moodle at the same time? If so, what version?
Are you using a non-standard theme? If so, which one and what version?
You could add this to your config.php to disable sesskey checking altogether:
$CFG->ignoresesskey = true;
... but it may leave you open to some types of security attacks. Not an ideal solution but may help in the short term to allow you in to troubleshoot further.
Or you can ignore people who have been doing this for a very long time and go with WL’s insightful advice... 😏