Some more info about server please ...
Operating System? Moodle version is?
Are you using DB for session or files?
Add to config.php file: $CFG->dbsessions='0';
That forces to files ... moodledata/sessions/
IF OS is linux and you have shell access to server, there are some command line only PHP scripts in quite a few places of moodle code, as well as task that can be executed from command line.
Plus there is the setup of cron.
If you have ssh access, login as root user then cd /path/to/moodlecode/admin/cli/
Then issue: php cron.php
IF the php finds php-cli same version that your moodle web is running, the above command will run cron and the ad-hoc task ... there are a couple related to sessions and tokens.
Also, from Section 7 of config-dist.php file, there are debug lines for moodle that you can copy into config.php to see if we can't get more info on the issue.
'SoS', Ken