Transfering Android App Data...HELP Please

I've recently managed to drop my Android tablet on the lovely tiled kitchen floor. Needless to say it's screen is now pretty stuffed now. I managed to convince my wife to let me buy a replacement, and most of my games data have a username etc, that I have managed to transfer my saves from one device to the other.

However Marvel Avengers Alliance doesn't have that facility on the do this. Does anyone know of anyway I can transfer the info from the old device to the new one, rather than having to start all over again.