Original sin - Are we all born with inborn sin from Adam?

Are babies entirely innocent? And then they sin, and this is their first experience of sin?
 
Romans 3:19  Now we know that whatever the law says, it says to those who are under the law, so that every mouth may be silenced...
 
How is every mouth silenced, if some are not chargeable with sin?
 
Romans 3:19  and the whole world held accountable to God.
 
This includes even infants, so all must be accountable to God, everyone must be.
 
Psalm 51:5 Surely I was sinful at birth, sinful from the time my mother conceived me.
 
This would indicate that sin was present, from the start.
 
Romans 3:10 As it is written: "There is no one righteous, not even one..."
 
Not even the youngest? Apparently not, and though Paul also quotes that "all have gone astray" (v. 12), this need not imply that all started out perfect because of some later straying.
 
And if "Christ died for all" (2 Cor. 5:14), that would also show that all needed redemption, again, even the youngest.

"Indeed, through the first Adam, we offended God by not observing His command. Through the second Adam, however, we are reconciled, and are made obedient even unto death. For we were debtors to none other except to Him, whose commandment we transgressed at the beginning." (Irenaeus, Against Heresies 5:16:3)
 
"And if we are all made to live in Christ as we were made to die in Adam, then, as in the flesh we were made to die in Adam, so also in the flesh are we made to live in Christ. Otherwise, if the coming to life in Christ were not to take place in that same substance in which we die in Adam, the parallel were imperfect." (Tertullian, Against Marcion 5:9:5, c. 210 AD)
 
"Everyone in the world falls prostrate under sin. And it is the Lord who sets up those who are cast down and who sustains all who are falling [Psalm 145:14]. In Adam all die, and thus the world falls prostrate and requires to be set up again, so that in Christ all may be made to live [1 Cor 15:22]." (Origen, Homilies on Jeremias 8:1)