If the options aren’t saving, the problem is most likely that the page used to have a page template with an old theme. Now that the theme has changed, and the page template no longer exists, WordPress is ignoring the saved values.
To verify this, try creating a new page and selecting the option. If it saves, that’s your problem.
The solution is to create a new page that’s identical to your homepage, but a “fresh start”. Then delete the old homepage.