Probably the story of Enkidu has the most parallels to the Old Testament (if we are talking about the story of Adam and Even in Genesis). Enkidu is the most uncivilized of men, and, to the civilized Mesopotamians, was on the level of an animal. He had hair all over his body, wore no clothing (this is significant), and didn't know how to eat bread. He is brought into civilization by a temple courtesan (a kind of publicly-sanctioned prostitute, as far as we can tell). She takes Enkidu out of his "state of nature", tells him he can no longer run wild with the beasts, and he must come and learn to live and speak in a civilized fashion. Enkidu makes the change from wild person to civilized person, but he does experience some longing for his old ways ("running with the beasts"). There are consequences, too, for his transition; Enkidu, because he helped his friend Gilgamesh (whom he wouldn't have helped if he had remained "wild") to fight the Bull of Heaven, Enkidu is condemned to death by the gods.
Enkidu's original state-of-nature existence does have elements of innocence to it. He knows nothing of the gods, he is sexually innocent, and he is free from the jealousies and politicking of the urban environment. His civilization brings him gifts, too, but the price of leaving his innocent, wild ways behind him is the displeasure of the gods.
Compared to the story of Adam and Eve, this story is not so fraught with morality; but there are several parallels. Adam and Eve, before the Fall, are naked (as Enkidu was). The offering of the apple is full of sexual undertones, as the approach to Enkidu by the courtesan was. There is no way that Adam and Eve can live off the plenty of the Garden after the fall -- they know too much, and are made to work for their food ever after. Enkidu, too, is brought into the urban environment of the angry gods and centralized food production. The paradisaical state the Adam and Eve and Enkidu lived in before,in which all their needs were taken care of, is lost when they become "knowledgeable" and civilized.
While these stories do have important differences, they both are myths about how people converted from nomadic, hunter-gatherer lifestyles to settled, agrarian economies in the Middle East sometime after 8000 BC. While Ekidu is much more of a wild man and a hero than Adam, and Adam is the first man while Enkidu is a man outside of a populous city, they are still representative of the first settlers who changed their environment and their ways enough to convert to civilized, settled society. That there would be losses in the change from one lifestyle to another -- especially one so drastic and history-altering -- is a given, and these stories deal with those losses in a metaphorical way that the cultures that produced them could understand