I also play on Xbox and am having a similar problem,l. I'm trying to set up a mining base on Mun so I need to send a satellite first. I have done this 5 times and each time I make sure to quick save as I go and when I'm done, go back to the space centre save the game there. If I then quit back to the main menu, and then re-enter the save, everything is where I left it and has been saved, but once I go to the dashboard and close the game/turn the Xbox off, the game goes back and my progress is lost.
I have tryed "cleaning" up my game as suggested, leaving me with only 5 flights in progress and this being my only Save file, yet the game still does not save. I'm not having issues with crashes like other but it's still very annoyed and makes the game pointless to play.
Though I do not know how or why, but in showing a friend the game, I build a simple rocket which I launched and then landed, and that seemed to have saved. I do not know why but that saved and nothing else does. It may be because I never make a same myself and relied on auto save? But I don't know?
If anyone knows how to fix this it will be greatly appreciated. Or even if there is a way to copy a Save file to see if that would work?