We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
Keith Green put verses 10-12 into a song. New Testament uses the principles found here in the comparison between the Pharisee who looks up to heaven and recounts his many acts of service to God and the dude who looks down at his feet, beats his chest, and says have mercy on a sinner like me. "For you have no delight in sacrifice; if I were to give a burnt offering, you would not be pleased. The sacrifice acceptable to God is a broken spirit; a broken and contrite heart, O God, you will not despise."
Create in me a clean heart, Oh God, and renew a right spirit within me.
The New Testament also lops off the first two verses, which provide the context for the psalm:
1. For the choirmaster, a song of David.
2. When Nathan the prophet came to him (rebuked him) because he slept with Bathsheba.
You are starting with verse 3 of the Hebrew original.
Some random translation notes (using your verse numbers to avoid confusion):
1. "have mercy on me" - the original Hebrew is "pardon me" or "deal graciously with me" - and "steadfast love" is the Hebrew word usually translated as "lovingkindness". Not much of 'steadfastness' in the original - the entire first phrase is about unearned pardon based on (fleeting) pity.
2. "Wash me thoroughly" uses a rare word in Hebrew related to the word for "dry land" in the Noah story - implying a landscape "scrubbed" and featureless. This is paired with the verb meaning "launder". So I'd render this as "scrub me clean"
The second phrase is "and purify me from my sin" - intensifying the first strophe.
4. "blameless when you pass judgement" is literally "meritorious in your judgement" - paralleling the first phrase. "Blameless" is a bit perplexing in context.
5. Better as: For I was formed in error, in sin my mother formed me"
The word I've translated as "formed" is very rare - basically using the word for "wineskin" as a verb. The womb and/or the growing fetus as an expanding wineskin.
Both 5 and 6 start with the same word - "for" or "behold" - so there's an implied parallel here:
For error is part of my human nature,
For You desire inner truth.
6. "Inward being" is literally "the plastered over (places)" - one's inner thoughts, unknown to others. This fits when you understand that the context is the sin with BathSheba.
Similarly "in my secret heart" is simply "in (that which is) sealed" - it's one word. The Hebrew original is always more compact, direct, and allusive.
"therefore" does not exist in the original. "You desire truth, You teach me wisdom."
7. This is the second time the translator has rendered "clean" instead of "pure".
The word for "purge" is, oddly, a verb based on the same root as "sin" - but it means to expiate sin. In modern Hebrew it is used to mean "sterilize". Hyssop was a medicinal herb back then.
So I'd render "purge me with hyssop and I will be purified."
8. "Let me hear" misses the causative verb form - not allowing, but making me hear. In other words, guide me in repentance.
"bones you crushed" is possible, but "bones you brought low" is more likely.
10. "New and right spirit" - in Hebrew "new" is the verb: renew in me an honest spirit.
11. "Your holy spirit" is literally "the spirit you sanctified" - the speaker's soul.
12. A "willing" spirit is a "generous" spirit in the original.
13. No "then" in the original. "I shall teach..."
16. The translator skipped the most interesting word in this verse:
"You do not want sacrifice or donation, a burnt offering pleases you not."
"Donation" is an odd word based on the verb form "I shall give". Obviously an early formula of proclaiming a vow.
The Hebrew reader immediately connects this to the most prominent use of this odd word - in the Torah:
Do not bring a harlot's donation or the wages of dog-fights to the house of Your Lord.
So this is not a negation of the Temple service - David longed to build the Temple - but of insincere, outward repentance (continuing the inner/outer theme).