General :: Backup App Data With Unrooted Phone?

I had a Nexus S. Got some games and got pretty far in them. I did not root the phone. Now I have an N4. I want to pick up where I left off on the new phone with regards to the games.. let's say, They Need To Be Fed as an example.

Now, neither phone is rooted, so obviously this complicates things, but it is not impossible. I can easily use Carbon to backup my app data and restore it onto the new phone without requiring a root. However I want to explore alternative methods.

So in a perfect situation, I would like the app data (and/or the app itself) to be backed up into one nice file that can be saved onto the new phone and then opened up to put the saved data where it needs to be. I'm not sure, but do .apk's contain app data as well? Could I save my current progress on TNTBF along with the app itself into a .apk and then extract the file on my new phone to continue where I left off?

What Carbon does is create a folder specific for the app and its data, which is a bunch of files I'm not familiar with, and requires Carbon to restore the data itself. Although this app is phenomenal and will be my last resort, I'd like something that doesn't require the app itself to restore the data. I used to use something called Ultimate Backup I think. When I selected to backup an app, it would save a .apk to the sd card. I'm not sure if it was backing up just the app or the app data as well, but if it did save the app data then that would be optimal.