I have had this problem every year for the past several years using a wildcard certificate from GoDaddy. It always takes me a day set aside to fix it.
The issue I have had on my server is that when the new certificate is installed, and you try to switch Open Directory from the old certificate to the new one, the configuration never changes over. OD continues to look for the old cert even if you delete it from /etc/certificates.
My server has the correct name, DNS is set up, and all that stuff...So here's what I did this time that worked, after the new certificate was installed:
1) Open Server.app
2) Go to Server > Certificates and using the Secure Services using: dropdown, select Custom...
3) Set the Open Directory service to use "None" as the certificate and click OK
4) Go to Advanced > Open Directory
5) Turn off Open Directory (wait for it to shut down)
6) Turn Open Directory back on
7) Go to Logs and check the Open Directory Configuration log to make sure there were no errors on startup
8) Go back to Server > Certificates and using the Secure Services using: dropdown, select Custom...
9) Set the Open Directory service to use your new SSL certificate - in this case, it's my GoDaddy certificate and click OK
After doing this and waiting a few moments...
10) Go to Server > Certificates and using the Secure Services using: dropdown, select Custom...
...and if successful you will see your new certificate selected. I would also check the OD Configuration log again to make sure.