If I read it in a "trustworthy" news source (for me this is newspapers like New York Times, Washington Post, etc), I know that these institutions have a reputation to loose which incentivizes them to produce quality journalism.
If the New York Times started to spread AI generated false information or other content that I would deem low quality, I would switch to other news sources without those flaws. If there is no news source left that produces quality journalism and has a reputation for it AND there is nobody who cares about such journalism being produced then we have bigger problems. Otherwise, as long as there's demand, somebody will produce quality journalism, build a reputation for it and have incentives to keep not spreading false information.
Facts can be verified the same way they are right now. By reputation and reporting by trusted sources with eyes on the ground and verifiable evidence.
Regarding comments on news sites being spammed by AI: there are great ways to prove you are human already. You can do this using physical objects (think Yubikeys). I don't see any problems that would fundamentally break Captchas in the near future, although they will need to evolve like they always have.