The amount of people who did not understand the OP is hilarious. He tried playing the current version of the game (1.3) on his friend's 360, which did not have any patches at all. So his save was from a later version of the game, so there would be no way that backwards-flying dragon could exist in his friends game. The game save is not compatible. Simple as that.
Not so Gnome, he's stated he used his console at a friends not his friends console.
You can play Skyrim offline... there use to be a way to delete Obivion cache data during start up by holding certain buttons, maybe this is the same for Skyrim. So when you started the game maybe you accedently deleted the cache.
Thats only the game cache, not the system cache with the patch data.