What Scripture does say is that if Adam and Eve disobey God that they would die.
Scripture may not say it, but Trent does. (See Council of Trent - Session the Fifth - DECREE CONCERNING ORIGINAL SIN - Canon 1 & 2)
If any one does not confess that the first man, Adam, when he had transgressed the commandment of God in Paradise, immediately lost the holiness and justice wherein he had been constituted; and that he incurred, through the offence of that prevarication, the wrath and indignation of God, and consequently death, with which God had previously threatened him, and, together with death, captivity under his power who thenceforth had the empire of death, that is to say, the devil, and that the entire Adam, through that offence of prevarication, was changed, in body and soul, for the worse; let him be anathema.
If any one asserts, that the prevarication of Adam injured himself alone, and not his posterity; and that the holiness and justice, received of God, which he lost, he lost for himself alone, and not for us also; or that he, being defiled by the sin of disobedience, has only transfused death, and pains of the body, into the whole human race, but not sin also, which is the death of the soul; let him be anathema:--whereas he contradicts the apostle who says; By one man sin entered into the world, and by sin death, and so death passed upon all men, in whom all have sinned.
In canon 2, physical death and spiritual death are confirmed to be caused by Adam's prevarication.
It affects our souls.
How can it affect the soul but not the body? If Original Sin removes the powers, confirmed to be held by Adam in Trent, such as invulnerability and immortality, does this not change the body and soul?
It gets trickier. If Adam's soul is fundamentally different to our own, is it a different type of soul? Humans before Adam had a sensitive soul, humans after have a rational soul, but Adam had a Hyper-Rational-Anima-6000?