How to transfer app data between two different game center accounts?

Hello, I really need some help.

I have looked all over the web for ways to transfer apps between two different game center accounts and different devices. Several weeks ago I started playing Campus Life on my relative's Ipad 2, therefore spending many long hours to get to this point in the game.

The problem is I just got an Iphone 5 and I really would like to transfer the data onto it, so that I would not need to start all over again.

My grandma has an account (Game Center) onto which the data has been saved and I would like to have it stored onto mine. On some forum someone said to sign into the account which has the data on both devices and to then open up the game on the new device. It worked. Then I signed out and logged in with my own account- the game started from scratch. I panicked. I went back on the other account, and the data reappeared. I have been going back and forth, the game changing as many times. I thought it would ask to back up my grandma's data on mine but I was wrong. I can not seem to save the game onto my new account, and I have even tried to back it up to Itunes and then copy it onto my phone after having signed into my own account. And I don't know how to properly use Icloud. I am a mess. I am completely lost and have found no help on the web. (All people seem to have to deal with is sharing Apple IDs with different Game Center Accounts for the kids and transferring apps from one of their devices to another.)

I am so worried I am going to mess up and will lose the whole thing completely.

I have just started another game with my own account but since I am using my grandma's, the data is erased. It's one or the other but I definitely prefer saving Campus Life. I do not like the idea of having to only pick one.

I do not know any technical terms, so if you understand or even have the slightest clue about what I am talking about, I applaud you.

This seems like mission impossible, so if anyone can help in any way, I would be really grateful.

Thanks in advance for your time and thoughts!