Psalm 4
This psalm probably dates from David’s flight before Absalom, 2Sa 16:1-23.  Even though David had reason to lament in this psalm, he ended it well with these words, "I will lie down and sleep in peace, for you alone, O Lord make we dwell in safety."   
Great way for us to drift off to sleep each night.

Psalm 12
David says that the words of the Lord are "purified seven times".   The word “seven” in the Scriptures denotes a complete or perfect number.  The words of the Lord are “perfectly pure” or flawless.

Psalm 20
David reminds us again in this psalm about the "saving power of his right hand."
And who sits at the right hand of God, the Father?  Who has "saving power"?

Psalm 25
“Remember not the sins of my youth and my rebellious ways; according to your love remember me, for you are good, O Lord.”
Great response to those who say that during their youth they strayed from God. David is asking that God not remember his sins and rebellious ways, but to remember him. He is asking for a separation of himself from his sins. Hate the sin. Love the sinner.

Psalm 32
“Blessed is he whose transgressions are forgiven, whose sins are covered.”
In the Old Testament times sins were only covered by the sacrifices they offered. Jesus removes sin and cleanses the conscience
Heb 9:13-14
For if the blood of bulls and of goats and the ashes of a heifer sprinkling the unclean sanctifies to the purifying of the flesh, how much more shall the blood of Christ (who through the eternal Spirit offered Himself without spot to God) purge your conscience from dead works to serve the living God? 

Psalm 38
It sounds as if David is tormented by his sins and is suffering physically, mentally, emotionally, and spiritually.  And he knows what to do to remedy that torment.  He tells us in a previous psalm - 25.
"For the sake of your name, O Lord, forgive my iniquity, though it is great."  
“I said, ‘I will confess my transgressions to the Lord’ – and you forgave the guilt of my sin.”

But even better than knowing that he needs to confess his sins, David knows that he is blessed because his sins are forgiven. (see Psalm 32)  and we as Believers in Jesus know that too. (see Hebrews 9:13-14).  He purses our consciences and enables us to serve the living God.  Halleluiah!