Neither "in sin" not "into sin" is good translation. The correct interpretation of the verse is apparently "from conception, from the womb, I already sinned", or more colloquially, "from the get-go I sinned". The adverb "already" is implied after the prepositional phrases בעוון and בחטא but is omitted for brevity.
It is difficult to back-read "original sin" into the OT. In order to do so, you would need to:
- Ignore the fact that there is no explicit mention or allusion to it in the OT
- Ignore the fact that "original sin" is an abstract doctrine regarding the nature of man and sin whereas the OT was written in an age an culture that was not familiar with or concerned with this type of abstract thinking
- Ignore the fact that the OT is not about the individual's relationship with God but primarily about the relationship of God to a collective, Israel, to which the the idea of "original sin" doesn't apply
- Ignore specific references against transference of guilt such as Deuteronomy 24:16 and Jeremiah 31:29-30
- Ignore the numerous verses implying respect for parents in the OT. It would be unthinkable in the OT world for someone, particularly David, to impugn his parents at all and particularly not for the act of conceiving him. Note that in reference to Solomon, there is no reference to the sin of his father in Solomon's conception although there clearly was.
In Psalm 51 specifically you need to read verse 5 (Masoretic verse 7) completely against of the thematic context of the Psalm in order to read it as "born in sin and therefore a sinner by nature". This reading would be tantamount to a denial of the personal responsibility that this Psalm is about. The writer isn't saying "God, what do you want from me, I am a sinner by nature", on the contrary, the writer is taking responsibility for what he has done and asking to be purified from sin (next verse), a possibility that contradicts the idea of "original sin".
So what does verse 5 actually say? Fortunately the more responsible modern translators such as NIV, Christian Standard Bible, Contemporary English Version, Good News Translation, Holman Christian Standard Bible, NET Bible, GOD'S WORD Translation, and the NRSV recognize the problem of translating as "born in iniquity" in view of the post-OT novelty "original sin", and therefore they translate contextually to the effect of "Even from the time of conception, in the womb I was a sinner". It is clear from the context of the previous verses that the writer is confessing his own sins only, and in forceful terms, and that verse 5, the final verse of the confession, is an exaggeration that expresses the writers feeling of remorse for his sins. This is an allusion to the idea that the defining characteristics of individuals is sometimes apparent from conception and in the womb, as in the stories of the birth of Jacob and Peretz in the book of Genesis. In this verse, the writer implicitly but starkly contrasts his own character with those giants of previous generations.
The use of "from the womb" imagery is a recurring OT theme. Apart from the stories of Peretz, Jacob and Esau mentioned above, we see this theme also in the birth stories of Samson and Samuel.
Regarding the Masoretic Hebrew for this verse, the poetic style, being that it is poetic, allows for reading the verse either as "in sin my mother conceived me" or as "even when my mother conceived me I sinned". However, the context of this Psalm, the prevalence of the "from the womb" themes in the OT, and the respect given to parents and previous generations in the OT rule out the reading that implies that the writer is saying that his parents sinned in conceiving him or that previous generations sinned, or that the he is disavowing his own responsibility in any way.
Curiously, the idea of "original sin" was not exclusively Christian. It also appears in some of the medieval Jewish commentators on this verse, such as the Ibn Ezra, who sees in this verse a "hint" that the author is referring to the fact that Eve only gave birth after she sinned. There is no parallel idea in Islam.