Are all sins cursed by their own nature?

The answer depends on whether one is reading the Book of Proverbs or the Book of Ecclesiastes. I realize this statement doesn't exactly take the bull by the horns, but the Bible is ambiguous on this question.

Passages implying "Yes"

The idea that sin carries its own curse can be found in Numbers 32:23

behold, you have sinned against the Lord; and be sure your sin will find you out.

In the New Testament we have sayings such as "the wages of sin is death" (Romans 6:23) implying that sin brings its own punishment.

In Proverbs 11:19 we see that

He who is steadfast in righteousness will live, but he who pursues evil will die.

Passages implying "No"

But in the Book of Ecclesiastes, the Preacher declares "All is vanity" and bemoans the fact that evildoers share the same fate as the righteous:

I have seen everything; there is a righteous man who perishes in his righteousness, and there is a wicked man who prolongs his life in his evil-doing. Be not righteous overmuch, and do not make yourself overwise; why should you destroy yourself? (Ecc. 7:15-17)

Even Jesus, perhaps surprisingly, echoed Ecclesiastes when taught that God "makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust."

Since the OP mentions sexual sins, we might consider the story of the Woman Taken in Adultery. She is not punished for her sin and Jesus sends lets her go with only an admonition to "sin no more." But certainly the woman faced consequences, for her sin was known and must have been reported to husband, not to mention issues related to a guilty conscience.

In the end, if good deeds are their "own reward" then the opposite is probably also true - that sins are their own punishment - at least in the long run. So the best answer is "yes," sins do carry their own "curse," the teaching of Ecclesiastes notwithstanding.

Sin does not have a nature, but humans have a sinful nature. Sin is described in the Bible as transgression of the law of God (1 John 3:4) and rebellion against God (Deuteronomy 9:7; Joshua 1:18). Sin has been passed down through all the generations of mankind and we, Adam’s descendants, have inherited sin from him. Romans 5:12 says that through Adam sin entered the world, and so death was passed on to all men because “the wages of sin is death” (Romans 6:23).

Through Adam the inherent inclination to sin entered into the human race, and human beings became sinners by nature. When Adam sinned, his inner nature was transformed by his sin of rebellion, which resulted in spiritual as well as physical death.

We are sinners not because we sin; rather, we sin because we are sinners. This passed-on depravity is known as inherited sin. Just as we inherit physical characteristics from our parents, we inherit our sinful natures from Adam. King David lamented this condition of fallen human nature in Psalm 51:5: “Surely I was sinful at birth, sinful from the time my mother conceived me.”

You mention various sinful actions, from “white lies” to sexual sins. Be assured, “Everything that does not come from faith is sin” (Romans 14:23). Yes, the effects of sin result in unhappiness and suffering, not just for the person who commits the sin, but also upon the recipients of sinful actions. Ultimately, the result of sin is sickness and physical death, but even worse than that, results in spiritual death.

Are Christians cursed because of sin? All who have come to saving faith in Christ Jesus have been justified by God because of their faith. This results in being at peace with God through Christ Jesus (Romans 5:1). There is no condemnation for those who are in Christ Jesus (Romans 8:1). God does not allow His children to be cursed. No one has the power to curse one whom God has decided to bless. God is the only One able to pronounce judgment.

God has won the victory, and in Christ we are overcomers. “Everyone born of God overcomes the world. This is the victory that has overcome the world, even our faith” (1 John 5:4). The shield God gives us “can extinguish all the flaming arrows of the evil one” (Ephesians 6:16). We are free to worship God without fear (John 8:36). “The Lord is my light and my salvation—whom shall I fear? The Lord is the stronghold of my life—of whom shall I be afraid?” (Psalm 27:1).

The first curse in the Bible was given by God. Death was the final consequence of Adam’s choice to sin. In Genesis 3:19, God tells Adam, “For dust you are, and to dust you shall return” (NKJV). For those who trust in Christ, though, the curse of death will be overcome (Ephesians 2:1–10). Rather than fear death, believers have the assurance of eternity that fuels the way we live.