This issue will not be resolved by simply citing Luke 3:38. We do not, in that text, have any indication whether that was a pre-lapsarian (pre-fall) or post-lapsarian relationship.

via Was Adam a Servant or a Son Before the Fall? || Feeding on Christ.

On the contrary, the issue is most certainly settled by Luke 3.38.  Here it is in context:

Now when all the people were baptized, and when Jesus also had been baptized and was praying, the heavens were opened, and the Holy Spirit descended on him in bodily form, like a dove; and a voice came from heaven, “You are my beloved Son; with you I am well pleased.”

Jesus, when he began his ministry, was about thirty years of age, being the son (as was supposed) of Joseph, the son of Heli, the son of Matthat, the son of Levi, the son of Melchi, the son of Jannai, the son of Joseph, the son of Mattathias, the son of Amos, the son of Nahum, the son of Esli, the son of Naggai, the son of Maath, the son of Mattathias, the son of Semein, the son of Josech, the son of Joda, the son of Joanan, the son of Rhesa, the son of Zerubbabel, the son of Shealtiel, the son of Neri, the son of Melchi, the son of Addi, the son of Cosam, the son of Elmadam, the son of Er, the son of Joshua, the son of Eliezer, the son of Jorim, the son of Matthat, the son of Levi, the son of Simeon, the son of Judah, the son of Joseph, the son of Jonam, the son of Eliakim, the son of Melea, the son of Menna, the son of Mattatha, the son of Nathan, the son of David, the son of Jesse, the son of Obed, the son of Boaz, the son of Sala, the son of Nahshon, the son of Amminadab, the son of Admin, the son of Arni, the son of Hezron, the son of Perez, the son of Judah, the son of Jacob, the son of Isaac, the son of Abraham, the son of Terah, the son of Nahor, the son of Serug, the son of Reu, the son of Peleg, the son of Eber, the son of Shelah, the son of Cainan, the son of Arphaxad, the son of Shem, the son of Noah, the son of Lamech, the son of Methuselah, the son of Enoch, the son of Jared, the son of Mahalaleel, the son of Cainan, the son of Enos, the son of Seth, the son of Adam, the son of God.

The chronology of when Adam might have been given the title of “son” (“whether that was a pre-lapsarian (pre-fall) or post-lapsarian relationship”) is irrelevant. The relationship is by virtue of origin. Seth is Adam’s son because he came from Adam and Adam is God’s son because he came from God.  In both cases they received their life from the other.  There is no suggestion here that there is some kind of “adoption.”  The issue is birth and origin. Adam was God’s son by virtue of God’s creation.  Adam after all was created in the image of God, and that is a mark of sonship: “When Adam had lived 130 years, he fathered a son in his own likeness, after his image, and named him Seth” (Genesis 5.3).

However, none of this means that we have to choose absolutely between son and servant.  There are times when those two relationships can be mutually exclusive, when a servant is not a son (as when the Prodigal Son asked to be let back in the home).  But Paul says that all children are like servants in their immaturity (Galatians 4.1).  Sons obey their fathers.

Nor does this term have any bearing at all on “the covenant of works.”  When Paul claims to be a bondservant of Christ he is not claiming to be in a covenant of works with Christ (Romans 1.1).  So tying the alleged status difference of son to servant has no bearing on anything.  It is confusion to even frame the question in this way.

Jesus was both the son of God and the servant of the Lord from the time (and before) he was born. Paul was both adopted by grace as a son and graciously called to be a servant of God.  Adam could be both as well.