Just to be pedantic, it is unclear if the Aztecs had "full writing". The notation system in the codices is probably more like a kind of mnemonic and accounting system along with a calendar. They probably couldn't write arbitrary sentences. But so little material survives we don't actually know!
The Maya though absolutely had "full writing" at least during the classical phase around 200 - 800 AD, and their writing probably derived from the older Olmec system. The Maya system worked much like classical Japanese or ancient Egyptian, and uses a mix of ideographs and purely phonetic symbols. They could, and did, write anything they could say. There are historical accounts of battles and names and etc. given in monumental inscriptions. And it is almost certain during the classical era that the Maya had full literature, with stories, mythology, histories, anthologies, religious scripture, etc. There were probably still priests who could read the Maya system when the Spanish arrived - one Jesuit recorded notes that indicate some of the phonetic values of some of the symbols correctly! - but they were mostly killed and most of the codices burned by the conquistadors.
It was not deciphered again until the 1970s and it's sadly still not widely appreciated that Mesoamerica had literate civilizations.
Fun fact: The Maya were such good astronomers and record-keepers that we have exact date alignment, which we have neither for Ancient Egypt or for Mesopotamia. For example any dates in Assyrian records are subject to a ~40 year window of uncertainty for exactly when they occurred. There is no such ambiguity for the Maya: Pacal the Great almost certainly died on August 29, 683.
The Maya though absolutely had "full writing" at least during the classical phase around 200 - 800 AD, and their writing probably derived from the older Olmec system. The Maya system worked much like classical Japanese or ancient Egyptian, and uses a mix of ideographs and purely phonetic symbols. They could, and did, write anything they could say. There are historical accounts of battles and names and etc. given in monumental inscriptions. And it is almost certain during the classical era that the Maya had full literature, with stories, mythology, histories, anthologies, religious scripture, etc. There were probably still priests who could read the Maya system when the Spanish arrived - one Jesuit recorded notes that indicate some of the phonetic values of some of the symbols correctly! - but they were mostly killed and most of the codices burned by the conquistadors.
It was not deciphered again until the 1970s and it's sadly still not widely appreciated that Mesoamerica had literate civilizations.
Fun fact: The Maya were such good astronomers and record-keepers that we have exact date alignment, which we have neither for Ancient Egypt or for Mesopotamia. For example any dates in Assyrian records are subject to a ~40 year window of uncertainty for exactly when they occurred. There is no such ambiguity for the Maya: Pacal the Great almost certainly died on August 29, 683.