It sounds incredibly frustrating to be stuck in that support loop, especially as a paying player! Dealing with technical issues in games, only to be blocked by unhelpful support processes, is a common pain. Have you considered looking for help on forums or community pages dedicated to the game? Sometimes other players have encountered similar issues and found workarounds or can point you to the correct place to find your Game ID. If you are looking for a different way to pass the time, you could try Omegle chat to connect with strangers online, but resolving your game issue is probably the priority.
