Did the New Testament writers know they were communicating the actual Word of God?
I believe they did. Let us consider the apostle Paul as an example. In 1 Corinthians 2:13 the apostle Paul said he spoke "not in words taught us by human wisdom but in words taught by the Spirit, expressing spiritual truths in spiritual words." In this passage, Paul (who wrote a huge portion of the New Testament) affirms that his words were authoritative because they were rooted not in fallible man but infallible God (the Holy Spirit). The Spirit here is the Spirit of truth who was promised to the apostles to teach and guide them into all the truth (see John 16:13).
Then, in 1 Corinthians 14:37, Paul says, "If anybody thinks he is a prophet or spiritually gifted, let him acknowledge that what I am writing to you is the Lord's command." Paul's writings have divine authority because they are "the Lord's command," not the words of fallen man.
In like manner, Paul says in 1 Thessalonians 2:13: "We also thank God continually because, when you received the word of God, which you heard from us, you accepted it not as the word of men, but as it actually is, the word of God, which is at work in you who believe." Again, the reason why Paul's words were authoritative is that they were rooted in God, not in man. God used Paul as His instrument to communicate His Word to man.
Regarding all this, note the following excellent summary formulated by the Synod of the Christian Reformed Church in 1961:
It is significant that Paul, while holding to the view of a canon of God-breathed writings, which constitute "the oracles of God," claims for his own teaching, either oral or written, equal status. The word that he preached was not "the word of men" but "the word of God" (1 Thessalonians 2:13). That which he wrote was "the commandment of the Lord" (1 Corinthians 14:37). He that does not obey the writing of the apostle is to be disciplined (2 Thessalonians 3:14). If any man preach or teach any other gospel than that which Paul proclaimed, he is to be accursed (Galatians 1:8-9). When Paul speaks as an apostle of Jesus Christ to the churches it is "Christ that speaketh" in him (2 Corinthians 13:3). Paul did not learn his gospel at the feet of men but rather received it "through revelations of Jesus Christ," Galatians 1:12, some of which were "exceeding great" (2 Corinthians 12:7).
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