It is clear that the new covenant believer is commanded to sing the Psalms as well as new songs.
Then you have two
prescriptions which cancel each other out. The contradiction is evident.
The only time two prescriptions necessarily contradict each other is when they are stated as "either/or". They do not contradict each other when they are stated as "both/and".
When the psalms command the singing of "new songs" they do so in a particular context. Ps. 98 gives us a clear statement of that context. The reason new songs were to be composed and sung in that period was that the Lord had "made known his salvation ... and his righteousness in the sight of the nations". A number of Psalms record such revelations of God's righteousness and judgements and the events mentioned are not merely pre-Sinai but cover the eras of the Exodus, Judges, Samuel to David (Ps. 78), David before becoming king (Ps. 34), various victories of Davidic period (Ps. 18), and even a pslam describing the Babylonian judgment (Ps. 137). Since the events covered include those contemporary with the writings of some of the psalms that describe them, it is clear that through the years from David to Babylon, the term "new song" could describe a newly composed psalm celebrating the Lord's ongoing deeds of salvation.
Yet the command to sing new songs is balanced by another command within Psalms, i.e., to "remember the wondrous works that he [the Lord] has done" (Ps. 105:5) and if older deliverances were already recorded in older psalms that meant those older psalms must be sung alongside and not instead of newer songs recording contemporary deliverances.
It is clear from the survival of the book of psalms to the era of the second temple that the pre-captivity Jews did not understand the exhortation to sing new songs to be an exhortation to sing new songs only, for they kept the older songs in their songbook. They understood that the instruction to sing new songs was a both/and instruction combined with a recognition that older songs that recorded God's earlier acts of deliverance remained valid expressions of praise and that singing them obeyed the instruction of Ps. 105:5.
Now the question we face today is not "Must we sing new songs only?" but "should we not sing new songs in addition to the biblical psalms? Why or why not?" Whatever the full intent of Paul's twice repeated instruction to sing "psalms, hymns, and spiritual songs" (Eph. 5 19, Col. 3:16), the biblical psalms are certainly included within that which Paul wants sung, since they are specifically identified by the term "psalms". What is at issue, and the point upon which the whole debate turns, is Paul's intended meanings for the other two terms "hymns" and "spiritual songs".
The term "spiritual songs" cannot be simply presumed to be another meaning for the psalms. During the years before the close of the canon, the Holy Spirit was operating in "charismatic" mode (prophecy, tongues etc.) and it seems that "hymns" (1 Cor. 14:26 ESV) were included within the exercise of these gifts Since that Paul's overarching theme in these verses is the regulation of charismatic gifts within orderly worship (1 Cor. 14:1-40). , it is not going to far to read the "hymn" here as equivalent to "spiritual song" in Eph 5 and Col 3. (If hymn is not held equivalent we have the phenomenon that Paul discusses a purely human presentation in the assembly when the rest of the phenomena discussed are almost certainly "charismatic"). Given this evidence we cannot exclude the possibility that Paul was referring to similar charismatic songs when he mentioned "spritual songs" in the Eph. and Col. verses. (I have pointed out elsewhere on the board that God giving new songs [whether charismatic or human composed makes no difference] to an allegedly EP only commanded church is enough on its own to disprove the EP thesis - although God has the right to temporarily allow something he has previously forbidden and will later forbid again, to be faithful to his nature, he must do so in a way that avoids contradicting himself. Unless he goes on record specifically allowing the forbidden practice for a temporary period and explicitly ends the period in which the temporary freedom is allowed, he necessarily creates confusion, contrary to his nature.)
Now if psalms means the biblical book of Psalms, and spiritual songs can mean Holy Spirit prompted songs in the undeniably "charismatic" era of the church, "hymns" can no longer necessarily be held to be another term for the biblical psalms over against its other known meaning, that of human composed texts. And if we cannot derive the conclusion that we are commanded to sing biblical Psalms only by necessary consequence from this text - (and against the evidence provided in this and the previous paragraph, we cannot do so) - then we must conclude that God has not required biblical Pslam only worship of the NT churches.