ATM the only way to move over your files from the Mazza Studio Version to the Grams Game one is doing it manually via file manager. It's currently not possible to do so on iOS but the devs are working on a way to move them as well.
You'll need some file manager program or do it through your computer.
- Go to this folder: android\data\com.mazzastudios.mergedragons and copy the folder files to a save place to make a backup
- Download the new Gram Games Version through the Google Play Store. This is a new game so you can actually have both versions on your phone/tablet.
- Start the game at least once to make sure the new user folder is created
- Copy the folder android\data\com.mazzastudios.mergedragons\files to the following folder: android\data\com.gramgames.mergedragons\ and say yes when it prompts you to overwrite the existing files.
- Start the Gram Games version and you should have all your progress.