In order to properly understand Romans 6 thru 8 it is important to notice that he consistently personifies "sin". Part of the way he does so is by the use of the definite article and part by anthropomorphism such as speaking of being "slaves to Mr. Sin". That is the case here:
NIV Romans 6: 6For we know that our old self was crucified with him so
that the body ruled by [Mr.] sin might be done away with, that we should no
longer be slaves to [Mr.] sin— 7because anyone who has died has been set
free [emancipated] from [Mr.] sin.
Once we recognize that it becomes easy to see that he is speaking of manumission:
I would translate: "he that has died has been emancipated from [Mr.] Sin".
Note that while Paul most often uses the word in relation to forensic guilt BDAG shows us that it has other usages. I have bolded usage #3 which is the relevant usage:
δικαιόω fut. δικαιώσω; 1 aor. ἐδικαίωσα. Pass.: 1 fut. δικαιωθήσομαι; 1 aor. ἐδικαιώθην, subj. δικαιωθῶ, ptc. δικαιωθείς; pf. δεδικαίωμαι Ro 6:7; 1 Cor 4:4; ptc. δεδικαιωμένος Lk 18:14 (Soph., Hdt.; Aristot., EN 1136a; et al.; pap, LXX; En 102:10; TestAbr A 13 p. 93, 14 [Stone p. 34]; Test12Patr; ApcSed, 14:8 p. 136, 15 Ja.; Jos., Ant. 17, 206; Just.; Ath., R. 53, 1; 65, 14) to practice δικαιοσύνη.
① to take up a legal cause, show justice, do justice, take up a cause τινά (Polyb. 3, 31, 9 ὑμᾶς δὲ αὐτοὺς … δικαιώσεσθε ‘you will (find it necessary to) take up your own cause’ = you will sit in judgment on yourselves; Cass. Dio 48, 46 ‘Antony was not taking Caesar’s side’ in the matter; 2 Km 15:4; Ps 81:3) δικαιῶσαι δίκαιον take up the cause of an upright pers. 1 Cl 16:12 (Is 53:11); τινί χήρᾳ (χήραν v.l.) 8:4 (Is 1:17 ‘take up the cause of the widow’).
② to render a favorable verdict, vindicate.
ⓐ as activity of humans justify, vindicate, treat as just (Appian, Liby. 17 §70; Gen 44:16; Sir 10:29; 13:22; 23:11 al.) θέλων δ. ἑαυτόν wishing to justify himself Lk 10:29; δ. ἑαυτὸν ἐνώπιόν τινος j. oneself before someone=‘you try to make out a good case for yourselves before the public’ 16:15 (δ. ἐαυτόν as En 102:10; but s. JJeremias, ZNW 38, ’39, 117f [against him SAalen, NTS 13, ’67, 1ff]). ὁ δικαιούμενός μοι the one who vindicates himself before (or against) me B 6:1 (cp. Is 50:8). τελῶναι ἐδικαίωσαν τὸν θεόν βαπτισθέντες tax-collectors affirmed God’s uprightness and got baptized i.e. by ruling in God’s favor they admitted that they were in the wrong and took a new direction (opp. τὴν βουλὴν τ. θεοῦ ἀθετεῖν) Lk 7:29 (cp. PsSol 2:15; 3:5; 8:7, 23; 9:2).
ⓑ of experience or activity of transcendent figures, esp. in relation to humans
α. of wisdom ἐδικαιώθη ἀπὸ τῶν τέκνων αὐτῆς is vindicated by her children (on δικ. ἀπό cp. Is 45:25. S. also Appian, Basil. 8: δικαιόω=consider someth. just or correct) Lk 7:35; also ἀπὸ τῶν ἔργων αὐτῆς Mt 11:19 (v.l. τέκνων). On this saying s. DVölter, NThT 8, 1919, 22–42; JBover, Biblica 6, 1925, 323–25; 463–65; M-JLagrange, ibid. 461–63. Of an angel Hm 5, 1, 7.
β. of God be found in the right, be free of charges (cp. TestAbr A 13 p. 93, 14 [Stone p. 34] ‘be vindicated’ in a trial by fire) Mt 12:37 (opp. καταδικάζειν). δεδικαιωμένος Lk 18:14; GJs 5:1; δεδικαιωμένη (Salome) 20:4 (not pap). Ac 13:39 (but s. 3 below); Rv 22:11 v.l; Dg 5:14.—Paul, who has influenced later wr. (cp. Iren. 3, 18, 7 [Harv. II 102, 2f]), uses the word almost exclusively of God’s judgment. As affirmative verdict Ro 2:13. Esp. of pers. δικαιοῦσθαι be acquitted, be pronounced and treated as righteous and thereby become δίκαιος, receive the divine gift of δικαιοσύνη through faith in Christ Jesus and apart from νόμος as a basis for evaluation (MSeifrid, Justification by Faith—The Origin and Development of a Central Pauline Theme ’92) 3:20 (Ps 142:2), 24, 28; 4:2; 5:1, 9; 1 Cor 4:4; Gal 2:16f (Ps 142:2); 3:11, 24; 5:4; Tit 3:7; Phil 3:12 v.l.; B 4:10; 15:7; IPhld 8:2; Dg 9:4; (w. ἁγιάζεσθαι) Hv 3, 9, 1. οὐ παρὰ τοῦτο δεδικαίωμαι I am not justified by this (after 1 Cor 4:4) IRo 5:1. ἵνα δικαιωθῇ σου ἡ σάρξ that your flesh (as the sinful part) may be acquitted Hs 5, 7, 1; δ. ἔργοις by (on the basis of) works, by what one does 1 Cl 30:3; cp. Js 2:21, 24f (ἔργον 1a and πίστις 2dδ); διʼ ἐαυτῶν δ. by oneself=as a result of one’s own accomplishments 1 Cl 32:4. (cp. κατὰ νόμον Hippol., Ref. 7, 34, 1).—Since Paul views God’s justifying action in close connection with the power of Christ’s resurrection, there is sometimes no clear distinction between the justifying action of acquittal and the gift of new life through the Holy Spirit as God’s activity in promoting uprightness in believers. Passages of this nature include Ro 3:26, 30; 4:5 (on δικαιοῦν τὸν ἀσεβῆ cp. the warning against accepting δῶρα to arrange acquittal Ex 23:7 and Is 5:23; δικαιούμενοι δωρεάν Ro 3:24 is therefore all the more pointed); 8:30, 33 (Is 50:8); Gal 3:8; Dg 9:5. For the view (held since Chrysostom) that δ. in these and other pass. means ‘make upright’ s. Goodsp., Probs. 143–46, JBL 73, ’54, 86–91.
③ to cause someone to be released from personal or institutional claims that are no longer to be considered pertinent or valid, make free/pure (the act. Ps 72:13) in our lit. pass. δικαιοῦμαι be set free, made pure ἀπό from (Sir 26:29; TestSim 6:1, both δικ. ἀπὸ [τῆς] ἁμαρτίας) ἀπὸ πάντων ὧν οὐκ ἠδυνήθητε ἐν νόμω Μωϋσέως δικαιωθῆναι from everything fr. which you could not be freed by the law of Moses Ac 13:38; cp. vs. 39. ὁ ἀποθανὼν δεδικαίωται ἀπὸ τ. ἁμαρτίας the one who died is freed fr. sin Ro 6:7 (s. KKuhn, ZNW 30, ’31, 305–10; EKlaar, ibid. 59, ’68, 131–34). In the context of 1 Cor 6:11 ἐδικαιώθητε means you have become pure.—In the language of the mystery religions (Rtzst., Mysterienrel.3 258ff) δικαιοῦσθαι refers to a radical inner change which the initiate experiences (Herm. Wr. 13, 9 χωρὶς γὰρ κρίσεως ἰδὲ πῶς τὴν ἀδικίαν ἐξήλασεν. ἐδικαιώθημεν, ὦ τέκνον, ἀδικίας ἀπούσης) and approaches the sense ‘become deified’. Some are inclined to find in 1 Ti 3:16 a similar use; but see under 4.
④ to demonstrate to be morally right, prove to be right, pass. of God is proved to be right Ro 3:4; 1 Cl 18:4 (both Ps 50:6). Of Christ 1 Ti 3:16.—Lit. s. on δικαιοσύνη 3c.—HRosman, Iustificare (δικαιοῦν) est verbum causativum: Verbum Domini 21, ’41, 144–47; NWatson, Δικ. in the LXX, JBL 79, ’60, 255–66; CCosgrove, JBL 106, ’87, 653–70.—DELG s.v. δίκη. M-M. EDNT. TW. Spicq.
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., pp. 249–250). Chicago: University of Chicago Press.