Nope. No one is responsible for anyone else's sins. The sins Jesus carried were ours, not His. He was completely without sin.
I'm not saying we are responsible for Adam's sin.
Therefore, just as sin entered the world through one man, and death through sin, so also death was passed on to all men, because all sinned. Rom 5:12
Adam sinned by eating the forbidden fruit and was sentenced to death. The death penalty has passed to all man because when Adam sinned we all sinned. All that means is that we share the same flesh as blood nature as Adam. We are all born in Adam and the nature he had when he sinned we all have. Therefore, the nature is what was sentenced to death and that same nature sinned in Adam. Adam represents all flesh and blood man.
Jesus shared the same nature that has been sentenced to die. Therefore, Jesus also sinned in Adam by sharing his nature. It's simply a matter of having the misfortune of being born in Adam.
Jesus overcame the sinful nature by personally committing no sin. What was condemned on the cross was that nature we all share being in Adam. The only way we can escape the death penalty of being in Adam is to be in Christ. Adam disobeyed God and brought death to all. Christ obeyed God and brings life to all in him.