DON'T SAVE YOUR GAME. Without saving, load your current save from the Pause Menu, which should bring you back to your previous manual save or latest autosave. (Make sure to select LOAD rather than SAVE.) You'll likely have to replay a few minutes of content to get back to where you were in the story when you got stuck.
However, if you have already saved your game while in a stuck state, or upon loading the game you are still stuck in the same spot, unfortunately our best recommendation right now is to restart the game. (We have to no way to repair save files on Xbox—we spent a while looking into this!) We know this isn't what you want to hear, and we wish we had a better answer.
Hopefully, you enjoy the rest of your time in Firewatch, and again, we apologize if you were affected by this issue.