I think changing the serverName and updating the alfresco-global.properties is enough. share-config-custom.xml file change is required in case your alfresco and share are running on different hosts. There is nothing to do with DB in this case.
I am not sure what is your current setup. But considering that you have a proxy host, a db host and a host where alfresco and share is deployed.
I would change:
Required so that proper URLs are prepared:
alfresco.port=80 #As per your current setup. This is an example
share.port=80 #As per your current setup. This is an example
If you are using CMIS to connect, you may also have to set some of the opencmis properties:
[Edit]: If share to alfresco communication is configured to via proxy, then share-custom-config.xml is also required to be modified.
(ACSCE, AWS SAA, Azure Admin)