This is not doable without reshaping of your data. Here is the idea section for this: - http://community.tableau.com/ideas/1498
Here is some good discussion on this: - Two independent "Measure Values" pills on dual axis chart?
Check these. Hope these may help you!
I have actually looked at those threads, but it wasn't clear to me that the re-shaping examples were directly applicable to my current need. In those examples, people were essentially making a single chart with multiple measures on each axis (i.e. one x-axis with a dualed y-axis). I need to have multiple x-axis, each of which is sharing a y-axis, and each with its own dual-y-axis variable.
At worst, I figure I can build each subplot individually and mash them together into a dashboard, but that obviously has some major drawbacks.
I do not think that this is possible. But your thought of putting them together in a single dashboard is fine & may be the only method. But may be someone from community may help into this.