How do justification and adoption relate? Look at adoption in the first century context, then see how Michael Horton describes it. This is helpful, in my opinion (not to mention quite comforting). I’ve put key terms in bold for ease of reference.
“Adoption was far more frequent and important in Roman society than it is today. The person adopted (at any age) was taken out of his previous condition, all old debts were cancelled, and he started a new life in the relation of sonship to the new paterfamilias, whose family name he took and to whose inheritance he was entitled…. Adoption was a legal act, attested by witnesses” (Everett Ferguson, Backgrounds of Early Christianity 3rd ed. [Grand Rapids: Eerdmans, 2003], 65-6).
Now Horton. “If union with Christ in the covenant of grace is the matrix for Paul’s ordo, justification remains its source, even for adoption. We do not move from the topic of justification to other…ones, but are always relating the riches of our inheritance to this decisive gift [justification]…. Once again we see that the antithesis between forensic and effective or legal and transformative is unwarranted. Adoption, like justification, is simultaneously legal and relational….”
Now hear this: “The tendency to replace the legal exchange with some notion of transfer of substance, properties, or habits in justification would have as its corollary a concept of adoption in which the adoptee, no longer adopted, receives a transfer of DNA” (Michael Horton, Covenant and Salvation [Louisville: Westminster John Knox, 2007], 247).
In sum, adoption in the biblical (ordo salutis) sense is legal and relational, as we can even see from Roman culture in the 1st century AD. At the same time, it is not based on anything inside a person (see my earlier post here on Covenantal Ontology) any more than an adoption today depends upon an actual change in the adoptee’s biological makeup, or DNA.
Justification is the source for adoption; both have to do with an external act/declaration of God, not something wrought in us. Side note: the Westminster Larger Catechism uses clear terms which agree with the above — adoption is an “act” (not “work”) of God (WLC 74).