How possible is it that the Septuagint / LXX wasn't specifically quoted in the NT? Couldn't it be possible that the NT writers quoted from a Hebrew or Aramaic Old Testament, and that the Septuagint was transliterated afterwards, or that the NT was later translated to match the Septuagint?

The reason why I ask this is because I can't really seem to find much proof of the age of the Septuagint. The two arguments for the authenticity of the Septuagint seem to be "the NT writers quoted from the Septuagint" and the anecdote of a Jew named Aristobulus, who mentioned that the Law had been translated into Greek under the reign of Ptolemy. Don't Jews refer specifically to the books of Moses as the Law? And wouldn't it be a big stretch to assume that he is truly speaking about the Septuagint being translated during the reign of Ptolemy? Yet this is what many advocate to be true.
It is highly unlikely. I will address the OP's commentary/questions in turn:


  The reason why I ask this is because I can't really seem to find much proof of the age of the Septuagint. 


I offer the following pedantic quibble only because it is directly relevant to OP's concern: when considering the historical creation of the Greek Jewish scriptures, it is unhelpful to think of "the Septuagint" as a monolithic piece of literature. (This amusingly titled lecture makes the point well.) The "Old Greek" scriptures were translated over a period of 2-3 centuries, in various locations (Jobes) Still, it's a good question as to how we know when these translations were made. 


  The two arguments for the authenticity ....


I'm not sure what is meant but "authenticity" here. For the sake of this answer, I'm going to assume OP means "creation prior to the turn of the era".


  The two arguments for the authenticity of the Septuagint[1] seem to be "the NT writers quoted from the Septuagint"....


(Here this almost sounds like "authenticity" has some theological weight, but I'm going to proceed by assuming this is purely a historical question about the date of the LXX.) 

Right, they certainly did. The OP further suggests that the NT writers actually quoted from the Hebrew and either: 


[the NT] was later changed to match the LXX;

This requires a very low view of the manuscript tradition by which we receive the Greek New Testament. Given the strength of NT manuscript evidence beginning in the 2nd C. and the absence of any evidence of a systematic scribal effort to change the manuscripts in this way, this is highly unlikely.2 The fact that the NT authors frequently draw on aspects of the LXX quotes that are unique to the Greek text in their arguments makes this all the more implausible.

Take Luke's account of Peter's application of Psalm 16 to the resurrection in Acts 2:26: "... my flesh will also dwell in hope..." (ep’ elpidi). The hope theme is characteristic of the Greek Psalms , but the Hebrew reads "my flesh dwells securely" (lābeṭaḥ) (see Aejmelaeus). In order to project (retroject) the choice of ep’ elpidi into the LXX, we would have to believe that Luke made this odd translation choice in one verse, and someone subsequently mined the Greek Psalter for words translating forms of bṭḥ ("to be secure") and changed most but not all of them to forms of elpizo ("to hope"). It is much more plausible that this represents a special proclivity (or whatever -- see Aejmelaeus) of the Psalms translator, and Luke was dependent on the form of the Greek Psalm 15  (Heb/EVV 16) that was passed down to him.  

or
the LXX was written to match the NT.

Again, this would involve fabricating arguments in the NT out of thin air when they depend on readings not found in the Hebrew. More concretely, there is ample evidence that the Greek versions were created earlier. The OP cites Aristobulus who mentions the Greek law having been written during the reign of Ptolemy.  More famous (and probably earlier) is the entertaining and pseudonymous "Letter of Aristeas", dated by scholars to the 2nd C. BCE. It presents an elaborate story of the creation of the Greek "Law". The tale is widely considered to be a fabrication, possibly written to defend a particular form of the Greek text. Nonetheless, it attests to the existence of an established Greek text by the end of the 2nd C. BCE, i.e considerably prior to the NT. Other sources prior to the turn of the era which assume the existence of Greek translations include:


Eupolemos, a Jewish historian who wrote in the middle of the 2nd C. BCE and seems to have used a Greek version of Chronicles
the Greek text of Wisdom of Joshua ben Sira (a.k.a Sirach), dated ~132 BCE, which contains a prologue that makes reference to a translation of "the law, the prophets, and the rest of the books".
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Additionally, linguistic studies show that the Greek translations comprising the "Septuagint" most closely approximate a phase of Koine Greek attested spanning from the beginning of the third to the end of the second centuries BCE. See Dines (pp. 50ff.) for details. She draws on Lee, 1983.

Most concretely, we have manuscript evidence of the Old Greek translations pre-dating the NT.  Please see this answer on Hermeneutics.SE for the details. A handful of manuscripts are dated prior to the turn of the era, and those dated to the 1st C. CE can also not reasonably be expected to have been altered to conform to a NT canon that did not  yet exist as such. 


We can see, then, that it is unlikely that the Old Greek translations were substantially dependent on the New Testament. On the other hand, there may well be isolated instances of textual "problems" in the LXX that were introduced as "corrections" to agree with New Testament quotations. I'm aware of an argument by Karen Jobes (a respected LXX scholar) that the variation between the Greek and Hebrew of Psalm 40:7-9 were introduced by Christian scribes to conform to the text of Hebrews 10:5-7, which (she argues) was creatively adapted from a version of the Greek text (which is no longer extant) that looked more like the MT.3 This is an unusual scenario, though, and it is only possible in a text like the Psalter where there is no manuscript evidence in the pre-Christian era (in contrast to the Pentateuch, minor prophets). Some would also argue that Romans 3:13ff influenced the Greek text of Psalm 14:3 (LXX 13:3).

While I would be interested in learning about more instances like those described above, I think the evidence is sufficient (and certainly most Septuagint scholars think the evidence is sufficient) that the original Greek translations of the Hebrew scriptures pre-dated the New Testament. The New Testament authors were dependent on (quoted, referenced, and drew insight and inspiration from) these texts.



 Notes, references:  


1. In the strictest sense, "Septuagint"  refers only to the Pentateuch. I take it that the OP's interest is wider.
 


2. One would expect to see various stages of this, incomplete changes, more variants around these (hundreds) of verses that would need to be changed, etc. There was not a single, authoritative NT manuscript (or an authoritative manuscript of any book) available that could be changed definitively and then reproduced perfectly, thereby avoiding any traces of the emendation. I encourage you to read a book about NT textual criticism to frame your thinking on the plausibility of this. 
 


3. This is a somewhat complicated but (IMO) elegant argument. For other, I think more standard, explanations of this discrepancy, see How is the Septuagint interpretation of Psalms 40:6 reconciled with the Hebrew text? on Hermeneutics.SE.
  


The information in this answer is attested in the following introductory texts. While Jobes/Silva is my personal favorite, Dines has more material specifically bearing on the OP's questions/concerns:



Karen Jobes and Moises Silva. Invitation to the Septuagint Baker Academic, 2nd Ed., 2015.
   


Jennifer M. Dines The Septuagint (T&amp;T Clark, 2004).
   

 
I have also cited, in addition to linked material:


Anneli Aejmelaeus, Faith, Hope, and Interpretation: A Lexical and Syntactical Study of the Semantic Field of Hope in the Greek Psalter in Studies in the Hebrew Bible, Qumran and the Septuagint presented to Eugene Ulrich Ed. Peter W. Flint et al. Brill, 2006.