There are three passages left to consider in that thesis. Let's begin these with 2Tim 4:
1I charge you in the presence of God and of Christ Jesus, who is to judge the living and the dead, and by his appearing and his kingdom: 2preach the word; be ready in season and out of season; reprove, rebuke, and exhort, with complete patience and teaching. 3For the time is coming when people will not endure sound teaching, but having itching ears they will accumulate for themselves teachers to suit their own passions, 4and will turn away from listening to the truth and wander off into myths. 5As for you, always be sober-minded, endure suffering, do the work of an evangelist, fulfill your ministry.In this exhortation to Timothy, Paul continues what he has already begun in the previous letter: he tells Timothy that it is important that he preach the word -- but he tells him that for a specific reason, and thereby limits what he might mean by saying, generically, "preach the word". Paul here, as he did previously, makes a strict distinction between "the word" or "truth" and "not sound teaching" or "myth". He says that what Timothy is charged to do is to provide the truth in spite of people "turning away from listening to the truth" who instead "wander off into myths".
This view of truth is reiterate by Paul to Titus:
Titus 1:5 This is why I left you in Crete, so that you might put what remained into order, and appoint elders in every town as I directed you-- 6if anyone is above reproach, the husband of one wife, and his children are believers and not open to the charge of debauchery or insubordination. 7For an overseer, as God's steward, must be above reproach. He must not be arrogant or quick-tempered or a drunkard or violent or greedy for gain, 8but hospitable, a lover of good, self-controlled, upright, holy, and disciplined. 9He must hold firm to the trustworthy word as taught, so that he may be able to give instruction in sound doctrine and also to rebuke those who contradict it.
10For there are many who are insubordinate, empty talkers and deceivers, especially those of the circumcision party. 11They must be silenced, since they are upsetting whole families by teaching for shameful gain what they ought not to teach. 12One of the Cretans, a prophet of their own, said, "Cretans are always liars, evil beasts, lazy gluttons." 13This testimony is true. Therefore rebuke them sharply, that they may be sound in the faith, 14not devoting themselves to Jewish myths and the commands of people who turn away from the truth. 15To the pure, all things are pure, but to the defiled and unbelieving, nothing is pure; but both their minds and their consciences are defiled. 16They profess to know God, but they deny him by their works. They are detestable, disobedient, unfit for any good work.
Let's consider that a second. Here Paul is not advocating that he trained Titus to have a liberty with the teaching so as to frame it according to some contemporary trouble or circumstance: he is telling Titus "I sent you to Crete in order to make teachers who will hold to the trustworthy word as taught." That means that Titus has been taught something which was itself "held to" – Paul was steadfast or saying the course to give it to him. It means that the word itself is "trustworthy" or "reliable" and not subject to interpolations or randy reinterpretations. The word, it can be said, is not the a result of what these men would like it to say: it is itself the source of what they ought to be teaching.
And the opposite – the results Titus must strive against – is that there are some who will not do this but instead are "unfit for any good work". They "deny [God] by their works". And what do they teach? "Jewish myths" – stories or claims which are of no part of the Gospel. And let me be clear: being Jewish is not what is at issue here because, plainly, Paul was a Jew. What is at issue is that there are teachings from the Jewish beliefs of the time which are contrary to the Gospel inherent in Jesus Christ.
Paul was not an anti-semite, and neither am I: the question at-hand is whether Paul was willing or able to say that the Gospel has a truth value which is not merely "real" in some literary or metaphysical way, but that the Gospel consists of true things which are juxtaposed against false things – things with no good use and only harmful use.
But this distinction is not limited to Paul. Peter also makes this distinction:
2Pet 1:12Therefore I intend always to remind you of these qualities, though you know them and are established in the truth that you have. 13I think it right, as long as I am in this body, to stir you up by way of reminder, 14since I know that the putting off of my body will be soon, as our Lord Jesus Christ made clear to me. 15And I will make every effort so that after my departure you may be able at any time to recall these things.
16For we did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ, but we were eyewitnesses of his majesty. 17For when he received honor and glory from God the Father, and the voice was borne to him by the Majestic Glory, "This is my beloved Son, with whom I am well pleased," 18we ourselves heard this very voice borne from heaven, for we were with him on the holy mountain. 19And we have something more sure, the prophetic word, to which you will do well to pay attention as to a lamp shining in a dark place, until the day dawns and the morning star rises in your hearts, 20knowing this first of all, that no prophecy of Scripture comes from someone's own interpretation. 21For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit.
So what can we make of the claim that the NT was written to be "real" as opposed to "true"? Should we ask whether this kind of vaguery is supported by the admonitions of the writers themselves. Given that those writers themselves make the distinctions between true and false, historic and fictional, and hold to a difference between "steadfast words" and "myths which lead away from sound doctrine", it is hard to establish the view that they do not have firm epistemological categories by which they view the world.
Now, here's the thing: if these categories do, in fact, exist in the NT, why would we want other categories to describe the truth claims of Scripture? I think the answer is obvious: we want to find a way to dismiss the truth claims of some of Scripture -- to find a way to overlook or ignore some of the claims in order to avoid being held responsible for those claims as we proclaim and announce (and defend) our faith. Let's be honest: that simply will not do. Trying to find inventive ways to dismiss Scripture is simply a lack of confidence in its sufficiency to teach us what is necessary for our faith -- even, and especially, when we find ourselves at odds against the culture and other religious claims.