Adam and Eve didn’t have immortal souls. In fact, the typical idea of an “immortal soul” isn’t one that’s from the Bible (see the answer to ‘How is “soul” used in the Bible?‘).
Like our ancestors Adam and Eve (Gen. 2:17; 3:19), all people are mortal (1 Cor. 15:21-22). We live for the duration of our life (Ps. 90:10), choosing how to use our time (Eccl. 9:10; Rom. 2:7-18). At the end of our life we die and cease to exist, moldering into dust (Eccl. 9:5-6,10; 3:20; Ps. 90:3).
No one goes to heaven when they die, not even saints like David (Acts 2:34), holy people who share God’s desires and do his will (Acts 13:22).1 What faithful people look forward to is not an “immortal soul” being in heaven, but the resurrection (1 Cor. 15).
In the future, Jesus will return to the earth to set God’s kingdom up here (1 Cor. 15:23-26). Part of the process of setting up the kingdom on the earth will involve a judgement (Rom. 2:1-16; 14:10; Acts 17:31; 2 Cor. 5:10), when ‘many of those who sleep in the dust of the earth shall awake’ (Dan. 12:2), and Jesus will say to some of them, “Well done, good and faithful servant. You have been faithful over a little; I will set you over much. Enter into the joy of your master” (Mat. 25:23); but about others, he’ll say, “You wicked and slothful servant! …cast the worthless servant into the outer darkness” (v26,30). The people who are welcomed into the kingdom will receive eternal life (Rom. 2:7) — and it is only at this point that any human (other than Jesus) will have immortality.
It is God’s intention for the earth to be inhabited by people who will live righteously (Gen. 1:27-28; Ps. 115:16; Hab. 2:14; Mat. 5:5). In contrast to the false idea of “immortal souls” going away to heaven, the Bible teaching about the resurrection matches this intention perfectly.
So, perhaps in answer to the specific questions in the title above, Adam and Eve didn’t need “immortal souls”, but, if judged favorably when Jesus returns, they (along with the other faithful people throughout history) will be made immortal around the time of the resurrection. Then they will live forever in perfection on a perfect earth, praising God.
After Death — What? on christadelphia.org
1. The only person to go to heaven is Jesus, and that was after he was resurrected (Acts 1:11).