Michael, on the drop-down menu for your date slider filters, do you have the option to set them to show "Only relevant values"?
Yep I went into Edit Filters and verified that the 'Only relevant values' checkbox is checked:
OK, then that falls into the "expected behavior" category. The underlying filter encompasses all possible values for those dimensions, and as you point out, only a subset of those values is relevant for any given department. Tableau shows that with the shading behavior you describe.
The important takeaway is that the filter is a single dashboard entity. I believe you can swap out filters using the "swap & pop" trick the same you can with individual vizzes on a dashboard - it might be worth a try to see if you can make it happen. The downside is that you have to have an individual viz for each department that a user can select, and that selection filter will need to be a parameter to drive the swap & pop.
See this link for ideas: We made a video of Sheet Swapping and Legend/Filter Popping on a dashboard.