Stories existed long before writing. Some of the most ancient texts we know of likely had a long existence as oral accounts way before they were first penned. Oral traditions do rely on the faultiness of memory, yes, but I think they are much more accurate than many give them credit for. Especially if your society is organized around these histories, they would take every precaution possible to prohibit the corruption of information. Many probably trained their whole life to be oral tellers. One particularly interesting technique I discovered recently is that storytellers often skipped a generation (so a grandparent would tell their tale to their grandchild, skipping their child), which I think is really smart. You would have half as many steps for something to go wrong.
Many cultures far removed from the west still have strong oral traditions. It is fascinating how many cultures who had no contact with each other have shockingly similar stories about a global flood. 🙂
Anyway, here is some insight to the design process of a major player in ch4, of which storytelling plays a large role. He likely has been a teller for most of his life, and is respected by his people as such. He is their library. They would trust him much more than some joe schmoe telling the same story.