Need help with xbox original game saving

My name is Fernando I'm an independent consultant, I'm happy to help.

When you are playing a backwards compatible game on an Xbox console or in the cloud, save the game frequently. If you open another game or app in full screen without saving, you will lose your unsaved progress.
When you're ready to end your game session, you'll need to return to the game's main menu before exiting to have your game saved to the cloud.
Note Xbox 360 game progress can be saved to Xbox 360 and continued to Xbox One, Xbox Series X|S, or the cloud. For original Xbox games, saved games, whether originally played on an original Xbox console or Xbox 360, cannot be accessed on newer Xbox consoles.

Hope I helped and see you soon

Hi ProR2D2 - Have you checked to see if there is any corrupt data for the game by going to the XBOX 360 Dashboard > Settings > System > Storage > Hard Drive > Games and Apps > Lord of the Rings: The Third Age. Check in there if you see any yellow exclamation
points next to saves. It could mean that the saves have been corrupted. If for some reason you don't see any saves at all, but you have saves for other games it could just mean that the saves didn't successfully finish or have just disappeared. If that did
happen, unfortunately there is no way to recover those saves, and I thoroughly apologize for that.

Might be a long shot, but double check you didn't save your game to the cloud rather than your harddrive.

Forgot to mention there was a blackout because of a storm and my 360 violently was turned off. But I had saves that should remain?

It already does that. You can store games and data on anything up to 2 TB.