I should just add that ‘Septuagint’ is really just a convenient shorthand term for ‘early Greek translations of OT books’, but it’s not a completely accurate one at that.
Originally, the term ‘Septuagint’ just referred to a particular Greek translation of a particular section of the Old Testament: a translation of the Torah (Pentateuch) made around the 3rd century BC.
A legend (mostly fictional, but with bits of fact in it) from the 2nd century BC claims that the translation was made by seventy-two (sometimes rounded into seventy) Jewish scholars for the Greek King of Egypt, Ptolemy II Philadelphus (reigned 285-246 BC), which explains the name:
versio septuaginta /
metaphrasis tōn ebdomēkonta ‘the version of the seventy’.
Strictly speaking, ‘Septuagint’ just refers to this 3rd-century BC translation of the Torah into Greek. However, some early Christians began including later (non-related) translations of other Jewish works into Greek - and some original Jewish compositions in Greek - under the ‘Septuagint’ label, which explains why nowadays, ‘Septuagint’ is used interchangeably with ‘Greek Old Testament’.
Strictly speaking, what the term ‘Septuagint’ means could either be (1) in a strict sense, the earliest translation of the Torah into Greek (ca. late 3rd century BC) - with the earliest Greek versions of other OT books (3rd-1st century BC) being called “Old Greek” (OG); or (2) in a more looser way, as a catch-all reference to these earliest available Greek translations of Jewish (scriptural) books
as a whole, or even (3) Scriptural books in Greek that were available to and read the early Christians, with no care as to whether they were the oldest translations or not. Nowadays the more common sense in popular usage is somewhere midway between (2) and (3), with more emphasis on (3).
The problem with the shorthand ‘Septuagint’ to refer to the Greek OT (“The evangelists/Paul quoted the Septuagint,” etc.), however, is that using it gives an inaccurate image in people’s heads. One might imagine ‘Septuagint’ here being used in the same way as, say, the terms ‘King James Version’ or ‘RSV’. In other words, it could give the image of (1) an already-established canon of OT books, and (2) that this established canon was already in scroll form (i.e. all the inspired works are contained in a single scroll or codex, like in our modern Bibles).
It’s inaccurate since (1) there was no ‘Septuagint’ in the sense of Greek OT canon to speak of yet (as mentioned), and (2) it wasn’t until the 4th century that
Christians got around to putting all the inspired works in a single volume. Up until then (in fact, even after the 4th century), different documents were all contained on different scrolls or codices. (So, for example, one scroll for the Torah, one scroll for the Psalms, one scroll for the minor prophets, one scroll for Isaiah, etc.) And it’s not always the case that a person or a community would own each and every document: so it wouldn’t be unusual for someone to own a copy of Isaiah but not, say, Judges, or for a small country synagogue or a local Jewish community to have the Psalms and/or Deuteronomy but not, say, Ruth.
That’s why the question “Did the NT writers use the Septuagint” can have different answers, depending on what exactly one means by ‘Septuagint’. Are you asking whether they used Greek translations of the OT books? Or do you mean that they used the Greek Torah? Or do you mean that they used the earliest translations made of these books (the so-called ‘Old Greek’ versions)?