Yes, a library project is exactly what you want. Put all the functionality in a library, and have two very thin shells for each of your app types that make calls into the library. As you noticed, duplicating the code is error prone and at best just extra work you shouldn't have to do.
See: Managing Projects for details. That page even mentions your exact scenario:
If you are creating an application that exists in both free and paid versions. You move the part of the application that is common to both versions into a library project. The two dependent projects, with their different package names, will reference the library project and provide only the difference between the two application versions.