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Nobody responded to this question, but hopefully if someone else has this same problem they will come to this page. So, after playing around with it a little longer and scouring ideas on the internet, I came up with a solution. This is what it now looks like:
There are a few key pieces that make this work. First of all, it was necessary to create two string parameters that contained all the values for State. Then, choose "Show Parameter Control" for each. Next, we need to create a calculated field with the following calculation:
This is saying if the ACTUAL DIMENSION equals the NEWLY CREATED PARAMETER then return the desired measure (Sales). Duplicate this calculated field, rename the second to State 2 Sales and change the formula to point to the second parameter instead of the first.
Next, it was necessary to drag these measures next to each other on the columns and bring the dimension that you want to break down these measures by to the rows shelf. In my case, it was FICO buckets (Category from Sales-Superstore that I renamed).
Next, bring Measure Names to the filters and choose the two newly created fields that are in your columns: State 1 Sales and State 2 Sales. Then, bring measure names to the color shelf for each chart. Change the size of the bars of one chart to be a little skinnier than the other.
Finally, click on the axis of the bottom chart and choose Dual Axis. Then click again on the axis and choose to synchronize axis and remove the axis for one of the charts so that you only have one remaining.