Publish the form template as a content type and have each department form library add it as a content type. Then, when you update the form template, all libraries will get refreshed.
Three ways to route the form:
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SharePoint Workflow - copy item and then delete item
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Different submit data connections - add 6 rules with different conditions based on the dropdown values and different submits
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qRules allows for dynamically changing the url for one submit data connection (slightly better than 2 since you don't have to have 6 different submits)
Best of luck,
Patrick