Justice not served?
The wages of sin is death.
I can accept Christs death for me and live and justice is served.
If I do not accept Christs death for me, then I have to pay the wages of sin, death myself. In both cases justice is served.

Actually the permanent wages of sin is the 2nd death with the consequences of sin being the 1st death upon which almost all of us experience regardless of whether we ultimately experience the 1st resurrection to Eternal Life and not experience the wages of sin in the 2nd death.

In other words, Christ paid the permanent wages of sin for us in that He experienced the 2nd death for us.

Did He die just the second death and not the first, or did He die both?

He obviously also died the 1st death.

I like the way it is portrayed in the book, Desire of Ages.
"The wrath of God against sin, the terrible manifestation of His displeasure because of iniquity, filled the soul of His Son with consternation. All His life Christ had been publishing to a fallen world the good news of the Father's mercy and pardoning love. Salvation for the chief of sinners was His theme. But now with the terrible weight of guilt He bears, He cannot see the Father's reconciling face. The withdrawal of the divine countenance from the Saviour in this hour of supreme anguish pierced His heart with a sorrow that can never be fully understood by man. So great was this agony that His physical pain was hardly felt. {DA 753.1}
And in that dreadful hour Christ was not to be comforted with the Father's presence. He trod the wine press alone, and of the people there was none with Him. {DA 753.4}
The cross brings to us two great questions, Why and Where was God?
"Why have you forsaken me?" Why did the most loving and kind being in the entire universe hide His face from his own Son?
"For God so loved the world, that He gave His only Son."
Where was God? It is beautifully told in the lament of David.
"And the king was much moved, and went up to the chamber over the gate, and wept: and as he went, thus he said, O my son Absalom, my son, my son Absalom! would God I had died for thee, O Absalom, my son, my son!" I believe eternity will not be long enough for us to comprehend what it cost the Father to hide His face from His own Son in His darkest hour.
Jesus is the only being in the universe who will suffer the hopeless despair of the second death, for whom it is unfair. He did nothing to deserve it. The wicked chose to reject the gift, and they are like the wedding guest without a wedding garment, speechless. What can they say when one was freely provided?
Moses said "If you cannot save this rebellious people, then blot my name out." Moses said it, but Jesus experienced it. How can we ever doubt God's love when He did not even spare His only beloved Son, to restore us back to Him? There is no greater price He could have paid.

I think this would refer to the first death:

Christ entered the tomb that man might pass through the tomb and rise with a resurrection-life. He burst the fetters of the tomb, and over the rent sepulcher of Joseph He proclaimed, "I am the resurrection and the life." And when the last trump shall sound, the Lifegiver will open the prison houses and those who have fallen asleep in Christ will come forth to a glorious immortality. {2SAT 112.4}

About the second death, EGW says the following:

But those who have not, through repentance and faith, secured pardon, must receive the penalty of transgression,—“the wages of sin.” They suffer punishment varying in duration and intensity, “according to their works,” but finally ending in the second death. ... Covered with infamy, they sink into hopeless, eternal oblivion. {GC 544.2}

Christ did suffer the punishment due to our sins, but did He die the second death?

Yes, He did. There is no terror in the first death. What is the punishment for our sins? To be cut off from the presence of God
with no hope of a resurrection. That is what Christ experienced when He uttered the anguished cry, "My God, My God, why have you
forsaken Me." The punishment or rather the wages our sins earn is seperation from God which results in death. The anguish of seperation broke the heart of the Son of God.
It is interesting that GC 544.2 says that the unrepentant must receive the penalty of transgression, - the wages of sin....finally ending in the second death.
That is the death that the wages of sin earns, if Christ did not pay that price, then He did not pay the full price for my sin. But praise God, on the cross He said "It is finished"
the price is paid in full, there is nothing owing.

Because "He could not see through the portals of the tomb,"
He experienced the hopeless despair of the second death. Sinners do not experience eternity in oblivion, only the pain of sinking into such a hopeless state. If you are wiped out (sunk in hopeless oblivion)you do not experience anything because you cease to exist.

Christ "brought life and immortality to light through the gospel" (2 Timothy 1:10). No man can have an independent spiritual life apart from Him. The sinner is not immortal; for God has said, "The soul that sinneth, it shall die" (Ezekiel 18:4). This means all that it expresses. It reaches farther than the death which is common to all, it means the second death. 1 SM298

{4SP 364.1}
"The wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord." [Romans 6:23.] While life is the inheritance of the righteous, death is the portion of the wicked. The penalty threatened is not merely temporal death, for all must suffer this. It is the second death, the opposite of everlasting life."

{ST, July 28, 1887 par. 13}
Christ has appointed to every man his work. The second death will be the portion of those who labor not, and the dreadful words will be heard, "Depart from me, ye that work iniquity." But the faithful servants will not lose their reward.

Just a few quotes illustrating the link between the second death and the wages of sin and departure or separation from God.

Yes, Christ did experience the terror of the separation from God, but did He die experiencing this terror? It seems there is a difference between what happened on the cross and what will happen with the wicked at the second death:

Suddenly the gloom lifted from the cross, and in clear, trumpetlike tones, that seemed to resound throughout creation, Jesus cried, “It is finished.” “Father, into Thy hands I commend My spirit.” A light encircled the cross, and the face of the Saviour shone with a glory like the sun. He then bowed His head upon His breast, and died. Amid the awful darkness, apparently forsaken of God, Christ had drained the last dregs in the cup of human woe. In those dreadful hours He had relied upon the evidence of His Father’s acceptance heretofore given Him. He was acquainted with the character of His Father; He understood His justice, His mercy, and His great love. By faith He rested in Him whom it had ever been His joy to obey. And as in submission He committed Himself to God, the sense of the loss of His Father’s favor was withdrawn. By faith, Christ was victor (DA 756).

Well said. Amen!