Depending on how big is your data, you can publish a packaged workbook (instead of publishing the data source and dashboard separately) to Site 1 and set a refresh schedule of daily. Then publish the same packaged workbook with the credentials of the other data source to Site 2 with a daily refresh.
I have a similar setup. But of course, it might not work for you depending on the details of your use case.
Thank you for your reply.
If I open my dashboard locally, it uses the data connection on Tableau server, not local. So if I try to export it to another Tableau site, it can't find that datasource. I think I may have caused this problem because the original dashboard became corrupted and I downloaded a copy from Tableau server. Also, I was under the impression that twbx files could not be refreshed automatically.
When you say "If I open my dashboard locally" are you saying that you open a .twbx/.twb file?
Package workbook published on a Tableau server can be refreshed by having a refresh schedule.