In my opinion, using published data sources for this suite of dashboards is absolutely the right approach to take - you're reducing duplication by centralising this.
I expect the issue you're seeing is due to caching.Have a read here Configure Data Connections and check your server settings. The only issue is this is a 'global' server setting, so you can't set these dashboards to 'not use cache' without affecting everything else on the server.
Alternatively, depending on how your dashboards get accessed by the users, you could add the ?:refresh=yes to the url which will force the page to load fresh data. This won't work though if you're reliant on users just navigating to your dashboard via the server UI itself. If you can control how they access the dashboard (eg via a link on a custom portal or similar then you can change the url).