I have already published an app called com.mycompany.mygame on google play.
I then decided to publish an ad free version of it. I did not change the package name in
As mentioned in other answers, you're development would be simpler if you put all the shared code and assets a common library project that is a dependency of your paid and free versions.
You may also wish to play with the new Gradle build system (in Android Studio) which allows you to dynamically set things like the package name at runtime. It also allows you to switch resources during build time, which is extremely convenient; You could have a boolean resource that represents whether the current app is the paid version. This allows you to enable/disable app features based on a check to that value.