When you say you get badge notifications do you mean that you only get badge notifications?
Are you not able to remember the steps you took to disable the notification?
If you have changed the settings back to allow notifications and you still are not getting them it could be that the app is buggy. There are 2 things I think you can try that might get you back up and running.
Since you are not able to uninstall the app, first try uninstalling the updates to Facebook. You can do this in the settings or at the Play Store page for the FB app. Settings>Apps>Facebook>Force stop>Uninstall. This will only uninstall the updates since FB is a pre-installed app that cannot be uninstalled fully without rooting the device. Uninstalling the updates shouldn't affect the app, so none of your settings should change. You might verify the notification settings afterwards just to be sure but I'm pretty confident they won't be affected.
I recommended this before, but you don't mention anything you've tried, so if you didn't then, the second thing would be to clear the app data. This should take the app back to default settings and will clear out any junk that might be causing the app to act up. Settings>Apps>Facebook>Force stop>Storage>Clear cache/Clear data. You don't have to clear the cache separately, it should clear when you clear the app data, but some users like to be extra and do both. If taking the app back to the absolute default doesn't resolve the issue this would be another clue that points to the issue being a setting or something on the phone and have nothing to do with the app.
Let us know what you find and we'll go from there.