'Radiation' is a word that conjures up images of Hiroshima, Chernobyl, cancer, mutations etc, and hence it's on that basis that moon hoax films and moon hoax books claim radiation would have been fatal to the Apollo astronauts. They know that the mere mention of radiation would put doubts in reader's minds, but it's a made up claim! You can see this for yourself by searching this book or any moon hoax book for the name (and quotes) from an astrophysicist or space radiation scientist from any nation (specifically those with no reason to support the USA) who says radiation would have made them ill, much less being fatal. You wouldn't find any, which should ring your alarm bells regarding the validity of such claims.
For example, here is what Dr Van Allen (the discoverer of the belts) said about such claims. Quote: "The recent Fox TV show, which I saw, is an ingenious and entertaining assemblage of nonsense. The claim that radiation exposure during the Apollo missions would have been fatal to the astronauts is only one example of such nonsense".
The main radiation problems for space travel are cosmic rays, which are constant, and solar particle events (SPEs) rather than solar flares, where SPEs are only a problem when headed our way and therefore are rare.
When cosmic rays collide with the atoms in a solid surface, they produce a spray of secondary particles which can be just as harmful as the cosmic rays themselves. This is what that NASA article was reporting. But you miss the other part of the 'story', which is when cosmic rays hit the hull of space craft and produce a spray of secondary particles inside the craft. Hence google search "Space station radiation shields 'disappointing'" and read the article of that title.
Quote: "Radiation inside the ISS, and the now defunct Mir, is caused when the fast, heavy ions that make up cosmic rays collide with the aluminium hull, releasing a shower of secondary particles into the living quarters".
So as you can see, a solid surface being hit by cosmic rays will result in sprays of secondary particle radiation, and therefore our proximity to that surface (whether it's the surface of the moon or the hull of a space craft) exposes us to that secondary radiation.
Where this becomes a problem is for long duration missions, like those to mars and back which would take at least 2 years with current rocket technology, and therefore either we need to find better shielding solutions (eg. design the hull of the craft as a tank to store the fuel, water, waste etc, which surrounds the astronauts) or develop rocket engines that will shorten the journey time significantly, or a combination of both.
I hope that addresses some of the concerns you had regarding radiation.