Can I turn off story progression for specific households?

As Katyfernlily mentioned, this is part of the point of NRaas StoryProgression, which can prevent pretty much every type of progression if you want. Check out @igazor 's rotational caste settings:

https://www.nraas.net/community/Story-Progression-Rotational-Caste-Settings

He's done the same thing it sounds like you want to for his protected households, so you could simply copy his settings into your own game. If you notice something you didn't want to happen, it's easy to adjust from there.