Open the .dtproj project file (it's just xml) and check what packages it is trying to open.
If you rename the package, you always get a message that asks you if you want to change the object name as well. Always answer Yes.
Furthermore, also change the GUID of the package when you copy it. You can do this by clicking on the control flow, hit F4, and look for the ID property. By clicking on that property, you can generate a new ID.