Unless I can present tablets? parchments or papyri, etc.? It's obvious to me at least that when they were compiled in their more or less present form they were taken from many centuries of copying.
No, you have only evidence of copying centuries after the Babylonian Exile, NOT BEFORE THEN.
We do have Babylonians copying the Epic of Gilgamesh (which included the flood myth of Utnapishtim) and the Epic of Atrahasis (another name for Utnapishtim) for centuries and for over 1400 years prior to the Exile. The Babylonians got their writings about Utnapishtim and Atrashasis from the older legend of Sumerian flood hero, Ziusudra (eg Eridu Genesis, as well as allusions in the Death of Bilgames and one of the Sumerian King Lists).
In 3rd millennium BCE Sumer (Early Bronze Age) and in 2nd millennium BCE (Middle & Late Bronze Ages) Babylonia, there were a number of scribe schools throughout Mesopotamia, where pupils learn writing by copying the Epic of Gilgamesh (or Bilgames), so just about every literate Sumerians and Babylonians knew well of Ziusudra/Atrahasis/Utnapishtim.
And during the Iron Age, Babylonians and Assyrians continued to copy many of the old myths (not just about Gilgamesh).
During the late 7th century BCE, the Neo-Assyrians even built a library at Nineveh, where the most famous tablets on the Epic of Gilgamesh were discovered by 19th century Assyriologists. These tablets may not be the oldest, but they contained the most complete version of the Flood story.
Ancient Mesopotamia have a very long history of literacy that spanned millennia, YoursTrue.
The Old Testament only dated to the Exile to -
- the Greek Septuagint,
- the Dead Sea Scrolls of Qumran caves,
- and the post-Temple (after 70 CE) Masoretic Text
...are the only main sources to the modern Bible translations.