Tubes do flicker which can contribute to the problem but I would want to confirm that it works in a particular case before buying a new television.
If your console is connected to the internet (there should be an ethernet jack in the back) and you’ve created a profile for XboxLive, once you sign in and start playing a game it should prompt you that there’s an update available. Hopefully you have a harddrive too- I’m not sure if you can install updates to a memory card.
for bioshock it should show up under options. it should say something with the words “FOV” turn it off or on. you should be able to see the effect it has right there on the options screen.