Hi @Max Johnson,
Thank you for posting in this community.
Backing up the content database is not enough if you want to backup and restore the search application from your old database to your new server. SharePoint's search application involves multiple databases.
Reference: Overview of search components and search databases
I recommend that you refer to the steps in this article to back up search service applications in SharePoint Server first: Back up Search service applications in SharePoint Server.
Then Restore Search service applications in SharePoint Server.
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