Here are some references from Encyclopedia Britannica:
On the Talmud
"Each of two groups of Jewish scholars (amoraim), one in Palestine and the other in Babylonia, independently produced a Talmud. Although the two groups addressed the same Mishna and consulted with one another, their work resulted in two separate collections of law, lore, and commentary. The amoraim of Palestine laboured for about two centuries, completing their work c.400 CE, approximately one century earlier than their counterparts in Babylonia. The Babylonian Talmud (Talmud Bavli) is consequently more extensive than the Palestinian Talmud (Talmud Yerushalmi) and, for that reason, more highly esteemed. Neither of the Talmuds covers every section of the Mishna; some commentaries were never written, and, presumably, others have been lost."
So whichever Talmud you were talking about it dates to many centuries after when Moses would have lived (which we do not even have a reliable date for).
On The Dead Sea Scrolls ("Biblical Literature and Its Critisal Interpretation")
"The importance of the Qumran scrolls cannot be exaggerated. Their great antiquity brings them close to the Old Testament period itself--from as early as 250-200 BCE. For the first time, Hebrew variant texts are extant and all known major text types are present.Some are close to the Septuagint, others to the Samaritan. On the other hand, many of the scrolls are practically identical with the Masoretic text, which thus takes this recension back in history to pre-Christian times. "
So Qumran shows a mix of texts.
Of the actual Masoretic texts
"No biblical manuscripts have survived from the six centuries that separate the latest of the Judaean Desert scrolls from the earliest of the Masoretic period. A "Codex Mugah," frequently referred to as an authority in the early 10th century, and the "Codex Hilleli," said to have been written c. 600 by Rabbi Hillel ben Moses ben Hillel, have both vanished. The earliest extant Hebrew Bible codex is the Cairo Prophets written and punctuated by Moses ben Asher in Tiberias (in Palestine) in 895. Next in age is the Leningrad Codex of the Latter Prophets dated to 916, which was not originally the work of Ben Asher, but its Babylonian pointing--i.e., vowel signs used for pronunciation purposes--was brought into line with the Tiberian Masoretic system."