So, my wife has been using her iPad for a while, I have been using it for a while. Now, we added both of us to a family account (with me being organizer). And, while I purchased an app which my wife had already purchased (before we set up the sharing), my assumption was that I wouldn't pay for it. But to my surprise, my card got charged. Is this the expected behavior?
It sounds like you went into the App Store > Searched for the App > Installed. Rather, with Family Sharing, you would want to navigate to the App Store > Tap your icon in the upper right corner > Purchased > from here you should see each member of the Family. Tap into one of them to view their purchases & ultimately download free of charge. When it is being downloaded free of charge you will see the iCloud icon with an arrow pointing down.
Hope that helps!