Fox affiliates don't have to push Fox News stories if they are not directly owned by Fox.

You're more likely to encounter forced programming from Sinclair than you are from Fox News.

Local stations sign affiliate contracts because they want non-news programming from the network. Local news is typically the biggest source of revenue for a station because they own all the commercial time. While the station owners will, from time to time, push a news story (looking at you Sinclair) most stations want as much editorial control as possible so they can cater to their local market and drive viewership.

So it's not impossible for a Fox News story to get pushed at the local level but it's not the status quo either.