39+ different answers
The Old Testament is not a single book, but a collection of 39 books (or 46, depending on who you ask).
Some of these books are well-attested in Hebrew before the 10th century, courtesy of the Dead Sea Scrolls. The Great Isaiah Scroll preserves almost the entire book of Isaiah and dates to the 2nd century BC. If we grant that the book of Isaiah was written in the late first-temple era (see Isaiah 6:1), that puts our earliest manuscript for Isaiah within about 600 years of the text's composition (that's actually not that long a timespan for an ancient document).
Other texts are not so well-attested at this early date; for example, the Book of Esther was not found at all in the Dead Sea Scrolls, putting our earliest copy of Esther closer to 1500 years after the text's composition.
The textual attestation of each book of the Old Testament (and in some cases, different portions of the same book) will vary based on the quality and quantity of manuscript evidence. Though the number of years is greater, this is not dissimilar to the textual transmission of the New Testament. There's a fragment of the Gospel of John that dates to approx. AD 125 (see P52 manuscript), only a few decades after original composition, whereas our earliest copy of 3 John is Codex Sinaiticus (see here), 2.5 to 3 centuries after the letter was composed.
Quality of textual transmission
The Great Isaiah scroll (from the Dead Sea Scrolls) has served as an effective way to gauge the accuracy with which the Masoretes preserved the text--as ancient documents go, they did a rather excellent job and were meticulous and careful. That said, there are approx. 2600 variants between the Great Isaiah scroll & the Masoretic text (see here), most of them quite minor.
The discipline of textual criticism seeks to reconstruct the original text based on the surviving manuscripts--it is not an entirely precise science, and scholars vary in their opinions, but it is generally acknowledged that for the NT, 95-99% of the autographical (original) text can be fully reconstructed from the manuscripts.
The figure for the Old Testament is subject to greater variation and debate, but would be somewhat lower. Nonetheless, the relatively small variation between the Old Testament texts we do have from the Dead Sea Scrolls & the Masoretic text speaks well for the Jewish scribes' careful preservation of their sacred texts.
Is the text we have today a 100% perfect copy of the original? No, but that is true of all ancient documents. What we do have is in remarkably good condition compared to other documents from antiquity.
New Testament use of LXX
The overwhelming majority of the New Testament's OT quotations are, as mentioned in the OP, from the Septuagint (The Gospel of Matthew is something of an exception, in that it draws from both the Greek & the Hebrew - I did a ~5-minute segment on this feature of Matthew in this video from 24:15-29:43).
This is what we would expect in documents written in Greek--the Septuagint was the accepted Greek rendering of the Tanakh to late 2nd-temple Jews. However, that does not mean early Christians did not know or did not use the Hebrew Tanakh. I make the case in this post that Jesus would indeed have been familiar with the Old Testament in Hebrew.
Whether or not the Masoretic text or the Septuagint more accurately preserves the original wording is a debate that will vary book by book and even passage by passage. The vicissitudes of textual transmission & textual reconstruction are such that there is no single blanket answer to this question.