It is true that most NT writers used one or more of the known forms of the Septuagint - or rather, the Old Greek. However, they do not always quote it - there are cases when writers will use a non-Septuagintal quote (for example, a translation from the Hebrew text - quite common in Matthew, for example), or a combination, or - when the extant texts do not meet their needs - hermeneutically alter the passage.
There are in fact different versions of the Septuagint. Limiting ourselves to the pre-Christian versions, there is, first and foremost, the
Old Greek (OG), which is the ‘original’. Between the original production of the LXX and the Greek versions of the 2nd century there were two early revisions. The earlier, dating from perhaps the 1st century BC, is called “proto-Lucian” because it shares characteristics with the revision attributed to Lucian of Antioch (4th century AD). It is called
proto-Lucian because some of the readings it shares with the later Lucianic recension are reflected in both Josephus (1st century) and the Vetus Latina (2nd century). This revision represents a sporadic correction of the Old Greek toward the Palestinian textual family represented by the three texts of Samuel found at Qumran.
The later revision is called the
Kaige after its habit of rendering the Hebrew word Hebrew ו)גם) “and also” as
kai ge. It dates to the late 1st century BC or the early 1st century AD and its overall character is seen as a revision of the Old Greek in the direction of conformity with the ancestor of the Masoretic text. The revision is noted for its use of stock renderings of particular Hebrew words and phrases (of which
kai ge is one) that seem to be chosen primarily to give Greek readers a sense of what was in the original Hebrew, with little concern for whether the resulting translation was idiomatic Greek. A Kaige text which shows close relation to another revision of the Septuagint attributed to
Theodotion (end of 2nd century) is called
Kaige-Theodotion (
Kaige-Th): the 1st-century
Minor Prophets scroll from
Naḥal Ḥever (8ḤevXII gr) is a Kaige-Th text. The extremely-literal Greek translation produced by Aquila of Sinope (ca. AD 140) was apparently influenced by Kaige-Th and is the fullest expression of the tendency to harmonize the Greek with the Hebrew. Theodotion’s version was somewhat later than Aquila’s, and his Greek is freer than Aquila’s stilted literalness. His religious affiliation is in doubt, but his version became popular with Christians, to the point that the rendering of Daniel attributed to him even came to replace the Old Greek version in the Church’s Greek OT.
The reason I brought this up is, whenever the book of Revelation cites from Daniel it always seems to quote Theodotion rather than the Old Greek. Since Revelation is usually dated to the last decade of the 1st century, you have the anomaly of a 1st century work citing a 2nd century version. Scholars have answered this by either proposing that Revelation uses a Kaige-Th version of Daniel or that the translation of Daniel attributed to Theodotion is actually not related to either proto-Theodotion or the traditional (2nd century AD) Theodotion, but is in fact a new translation of the Hebrew-Aramaic portions of the original book with some knowledge of Old Greek Daniel.