Personally, I keep all my played sims in My Households and keep the Stories off for them so things don't change when I'm not playing them. If you are keeping your households in the Other tab of Manage Households rather than in the My Households tab, you'll need to do it there as well.
You can also turn off just the adopt and have babies part of the Stories for all your sims too. You can turn the Stories off for all households in that tab or turn them off individually.
So, my suggestion is turn it off for played sims that you keep in the My Households tab in Manage Households. what you describe, it sounds like you have neighborhood stories on.Information, Guides and Announcements for the EA app.
