Or political records of what happened before the Europeans arrived? I understand this is a broad question. I just realized I knew that the Mayans and the Olmecs had writing, but knew nothing about what they were writing.
You may find this previous topic of interest.
In regards to your question regarding disease, perhaps the section Disease in the Americas from our FAQ will answer that inquiry.
That being said, the Olmec had very limited writing. In fact, we only know of one example of Olmec writing and that is the Cascajal block. The Cascajal block is covered in pictograms which may be an early form of writing, but is not a fully developed form of writing. The pictograms were probably used to a mnemonic device to recall memorized information in the same manner that the Aztecs used their pictographic writing. The later Epi-Olmec peoples, possibly descendants of the Middle Formative Olmec peoples, have what appears to be a fully developed writing system utilizing glyphs similar to Maya script or Zapotec script. Unfortunately too few lengthy examples of Epi-Olmec writing exists for a translation to be made. Attempts were made by John Justeson and Terrence Kaufman, but their proposed translation did not work when applied to a newly discovered mask containing Epi-Olmec script. One of the largest hurdles in deciphering Epi-Olmec is trying to determine that language in which the script was written. Hypotheses exist for particular languages, but none of them have been confirmed.
As for the Maya script, we have many examples of Maya writing. Writing appears on stone stela erected in the plazas of cities, on staircases or other decorative elements in architecture, on ceramic vessels, on murals, and on painted codices. The range of topics vary with Maya writing, but most of them tend of to focus on either the life and deeds of elites or mythology and religion. You do not necessarily see a diary of someone's thoughts, a treatise on a topic regarding morality, or a detailed description of how a pyramid was constructed. However, by analyzing the texts that are available and understanding modern Mayan languages, we can piece together some information that is not explicitly stated anywhere. One of the best books that covers topics regarding how the Maya saw themselves and the world around them is The Memory of Bones: Body, Being, and Experience Among the Classic Maya (2006) by Stephen Houston, David Stuart, and Karl Taube. It is heavy on linguistics and epigraphy, but I highly recommend it if these topics interest you. If you are more interested in the lives and deeds of elites there are a number of books on our books and resources list that are worth checking out and reading.
I know I'm very late to the party at this point, but I've only recently made an account and am going back through and revisiting a number of the posts that I've lurked through. The other two answers here are fantastic, I just want to chime in to answer your question in regards to South America. Although Quechua has persisted as a language, we don't have a ton predating the fall of the Incan empire in terms of recorded history, outside of a couple main sources. The first is by Inca Garcilaso de la Vega (not to be confused with the Golden Age poet with almost the same name), and is called "Comentarios Reales de los Incas". It's written in Spanish, and is pretty much the collection of a lot of the oral histories of the Inca's mixed with his own experiences growing up there before moving to Spain (he was born post conquest). There's a number of issues with the bias of the piece, for example his distortion of Incan religious practices, and he doesn't truly address a ton of the Spanish Conquest, more the ancient history of the Inca and how Cuzco is the equivalent of the Rome of the Americas. He does describe, however, that much of the history of the Inca is verbal, related through the female line (his text is based off of his interpretation of letters his friends in S. America sent him that they heard from their mothers who were alive before the Spanish Conquest).
Despite this, there does exist a "written" record of Incan culture, called the "Quipu" in English. I'd Google it for a quick picture, but it's essentially a woven strand that contains information. For a while they were thought to just be for accounting purposes, but that's recently been questioned (although there's no consensus yet). Modern written Quechua is pretty phonetic, and is fairly Spanish-oriented (a Spaniard made the first phonetic dictionary).