To review then...
On the IIS side, when you click the "Check In" button on a problem client, you see a corresponding log entry in the IIS logs for that client's IP address, and appears to be successful, correct?
You didn't confirm, but I'm assuming this was an inplace server upgrade using the same database; what version did you upgrade from?
On a problem client, if there are errors communicating, then the EAFrmCliComm log file under the Techlogs folder inside the software installation folder should be updated. Can you post this please?
Finally, the assignment of SEE clients to their OUs in the SEE Manager, is reliant upon the Directory Sync being up and running. Check your directory sync settings by opening the SEEMS Configuration Manager on your server and make sure it's all running and syncing.
On a final note, all of the above (client check-ins, directory syncing, etc) is dependent on the database being fully operational. Can you confirm the health of the database please (maintenance plans, backup strategy, available space, etc), and if has been moved as part of your upgrade (plus details please)?