Exactly! He is the model of us all. Struggling with his sin like all of us.
he trusted in the cleansing blood
Actually he didnt, Funny you should pick Ps 51.
16 You do not delight in sacrifice, or I would bring it; you do not take pleasure in burnt offerings.
17 The sacrifices of God are a broken spirit; a broken and contrite heart, O God, you will not despise.
And keep in mind, regarding David's sins, he intentionally killed Uriah, did he not? Animal sacrifices were primarily for UNintentional sins. Only a handful of intential sins are covered by Animal sacrifice: cheating your neighbor, bearing false witness and a few others. Murder is not one of them. Shed blood could not atone for his murder. How did King David obtain forgiveness? By using the above scripture. He confessed his sin with a broken and contrite heart. The prophet Nathan said HaShem has heard you and has ALREADY FORGIVEN YOU. No blood sacrifice.
I agree that he was aware of the messianic promise, through Solomon his Son, he would have NO cause to believe that the Mashiach ben David would be a hybrid god-man and that this god-man would sacrifice himself as a vicarious human atonement. David would have reviled such a thought as G-d in the form of a man.
“Wash me with Hyssop” is a direct reference to the blood of the Paschal lamb-