I am a bit lost on your use case. Why do you have a drop down for Month Filter? How will your user change the filter? AKA what is the work flow?
If you just want to show the last 6 months relative to today for both worksheets, then I recommend creating a relative date filter on each worksheet. While each worksheet will have a separate date filter using a different date field, both will be relative to today's date. For more information see Quick Start: Relative Date Filters - Tableau I have also attached an example.
If your use case is more complicated, then please provide some more information.
Hope this helps!