Open the AA installation directory, and inside that into the Server directory. In there you should find a file "ActiveAdministrator.xml".
Edit this file and you will see an xml fragment for each archive database. It will look like the example below. Please check the <ServerName> entries for the archive databases that are not working and ensure they are set to the correct server. If not, updating this to the correct server should correct the the issue.
Example fragment within the <Archives> section of the XML:
<Archive>
<ID>5a11421c-27d8-4e99-b806-a6186b96fe27</ID>
<Name>2019</Name>
<Active>False</Active>
<ServerName>MY-SQL-SERVER</ServerName>
<DatabaseName>ActiveAdministrator_Archive_2019</DatabaseName>
<IntegratedSecurity>True</IntegratedSecurity>
<ConnectionTimeout>15</ConnectionTimeout>
<EncryptConnection>True</EncryptConnection>
<TrustServerCertificate>True</TrustServerCertificate>
<Description></Description>
</Archive>