Where are the data sources stored? Is that location accessible from your Tableau Server? If Tableau Server cannot see the location where the PostGres files are stored at, it cannot update the extract as there will be a "connectionless failure". You will have to grant rights to the run-as user to be able to access the data. If the files are on a shared network drive that the run-as user on Tableau Server can access, make sure you are setting up the connection via the UNC path rather than the network drive letter.
More info here:
The data is stored in PostGres, but I've created extracts of it through Tableau Desktop that are saved on my computer. (Tableau won't let me publish PG datasources to the server with a "Live" connection, which is why I use extracts.)
The error message states that the tableau server is not able to sign in with the credentials embedded in the extract. I will suggest to click on "Change the password" for all the data connections and enter the password, and click on test option. If it successful, then try a full refresh of the data extract. I hope this will fix the issue.
Thanks @Kunal, but that did not work. I wonder if it's because some individual datasources in Tableau have multiple connections to the same PG database, which facilitated having filters on certain connections that I needed to organize my data.