Preparing this answer was my first time looking up how "original sin" is formally defined, and if I understand right, its definition may not be consistent across Christianity. So depending on your definition, the answer varies.
Do Latter-day Saints believe that we inherit some kind of intrinsic sin or guilt at birth as a result of the Fall of Adam?
No. You quote the second article of faith,
We believe that men will be punished for their own sins, and not for Adam’s transgression.
Additionally, from Moses 6:53-54,
53 ...And the Lord said unto Adam: Behold I have forgiven thee thy transgression in the Garden of Eden.
54 Hence came the saying abroad among the people, that the Son of God hath atoned for original guilt, wherein the sins of the parents cannot be answered upon the heads of the children, for they are whole from the foundation of the world.
Do Latter-day Saints believe that we inherit some kind of sinful state, or proclivity to sin, as a result of the Fall of Adam?
This one is also a no. From 2 Nephi 2:25-27,
25 Adam fell that men might be; and men are, that they might have joy.
26 And the Messiah cometh in the fulness of time, that he may redeem the children of men from the fall. And because that they are redeemed from the fall they have become free forever, knowing good from evil; to act for themselves and not to be acted upon, save it be by the punishment of the law at the great and last day, according to the commandments which God hath given.
27 Wherefore, men are free according to the flesh; and all things are given them which are expedient unto man. And they are free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil; for he seeketh that all men might be miserable like unto himself.
The redemption of Christ has made us free to choose whether to follow Him or to follow the devil. Now, this doesn't mean that we don't have weakness or other personal defects that lead us to sin, but I would argue from the scripture above that these of themselves are not effects of the Fall of Adam. Others may have additional perspective to give on this point, and I'd be happy to hear it and update my answer. There are certainly scriptures that talk about man becoming carnal/sinful by nature, but I'd still argue from the scripture above that those aren't direct effects of the Fall of Adam, at least to the degree that we avail ourselves of the redemption of Jesus Christ.
Do Latter-day Saints believe that there are negative effects of the Fall of Adam that, without the redemption of Christ, we would inherit by being born into this world?
Yes. The Fall of Adam brought about the separation of Adam and Eve and children from God. From Alma 42, various verses between 7 and 15,
7 And now, ye see by this that our first parents were cut off both temporally and spiritually from the presence of the Lord; and thus we see they became subjects to follow after their own will.
9 Therefore, as the soul could never die, and the fall had brought upon all mankind a spiritual death as well as a temporal, that is, they were cut off from the presence of the Lord, it was expedient that mankind should be reclaimed from this spiritual death.
14 And thus we see that all mankind were fallen, and they were in the grasp of justice; yea, the justice of God, which consigned them forever to be cut off from his presence.
15 And now, the plan of mercy could not be brought about except an atonement should be made; therefore God himself atoneth for the sins of the world, to bring about the plan of mercy, to appease the demands of justice, that God might be a perfect, just God, and a merciful God also.
The "temporal death" here is the separation of a person's spirit from their body, and "spiritual death" is the separation of a person from God. Both of these needed to be overcome for people to be able to return and dwell in the presence of God once again. Jesus Christ has provided redemption from temporal death free to all people without condition by one day resurrecting them all. Jesus Christ has provided redemption from spiritual death conditioned on obedience to the laws and ordinances of His gospel. But without Christ and the redemption He has made, mankind would forever be separated from God as a result of the Fall of Adam.
To return to Moroni 8:8 which is the basis of your question,
Listen to the words of Christ, your Redeemer, your Lord and your God. Behold, I came into the world not to call the righteous but sinners to repentance; the whole need no physician, but they that are sick; wherefore, little children are whole, for they are not capable of committing sin; wherefore the curse of Adam is taken from them in me, that it hath no power over them; and the law of circumcision is done away in me.
The curse of Adam is this separation from God, caused by a combination of both temporal (or physical) death and spiritual death. This definition is, I think, meant to be understood in context of other scriptures, particularly from the Book of Mormon and some of which I shared in my answer. At the very least, this teacher's manual for Moroni 8 says as much.
I would add as well, that in Moroni 8:8, the speaker (Mormon) indicates that this "wholeness" of little children is with them from birth. So even if the "curse of Adam" meant something akin to original sin, it couldn't even be that because the scripture indicates that this curse is taken away from little kids, and they are whole in Christ. This is the state they are born into, without any prerequisite (such as baptism), and they do not inherit any curse from Adam.
A good summary, as linked by @depperm, is the short Gospel Topics entry on original sin.