I find fault with your premise. Belief is not a choice, it is just something you have or don't have. Action is a choice.
As to the main issue, I don't think people willfully follow news they believe to be false. They might know the news source is biased but not care but I don't think they know it to be "fake news" Furthermore, "fake news" is being thrown around way too much, apparently in attempts to discredit other news agencies. Biased news, news reported by mistake, and news reported based on something believed to be true is not fake news. I would consider fake news something that is just absolutely made up. Anonymous sources are not "fake news." Anonymous source have been and continue to be real sources of information for journalists. Look at the anonymous source that helped break Watergate. A journalist needs to make sure they can trust an anonymous source and when reading a reader should consider that the source is anonymous, but an anonymous source does not make the information unreliable.
On top of that, the claim of bias has become akin to that of racism in that people think it to be some trump card where once you can prove bias then nothing that individual or agency reports every trustworthy. Someone having a bias doesn't mean they make up news or report false news. It might mean that their headline is less than neutral or what they choose to cover leans one direction but that doesn't mean we should just ignore things they report. Heck, the headline might not even be a bias but just shameful attempts to get readership. To be clear, I am talking about news stories, not opinion pieces or commentators.