Xbox one the cloud and game saves

Not exactly SW related, but I have beat the game. My old xbox one is having disc drive problems that are only getting worse. Upgraded to an xbox one s with 1 tb of hard drive space. 

Here is my question. Now that I am moving into an new xbox 1, I am trying to transfer my game saves between the two. I was told that I can only use the cloud to take my game saves with me. I did try to transfer my saves offline, but didn't work. I was told that was taken away from the gamers because of cheating. I would have to individually load my games into my old xbox and let them backup to the cloud. Then repeat the process on my new xbox to get them there. Is there a way to verify that the game saves are on the cloud? I don't have internet access at home and am trying to do this at work on breaks.