Rose,
There was no sacrifice under the law for adultery, only a sentence of death. The fact that David was granted mercy by God is indeed a type of the mercy extended to NT Believers. In Scripture it is called the 'sure mercies of David'.
Yes, David's adultery was premeditated. He planned it and used others in its execution. Hard to believe how far a person can fall, but no one is immune from it happening to himself. What is rarer still is repentance from such a fall.
In Psalm 51 David appealed to God's lovingkindness. In doing so he acknowledged his sin. This chapter details a repentance one could take lessons from.
Consider:
v.1: Have mercy upon me.
v.2: Wash me from mine iniquity, cleanse me from sin.
v.3: I acknowledge my sin.
v.4: Recognizes the seriousness of his sin
v. 7: Purge me and wash me.
v. 8: Make me to hear joy and gladness.
v. 10: Create in me a clean heart and renew a right spirit within me.
v.11: Do not cast me away from Your presence, do not take Your Holy Spirit from me.
v.12: Restore to me the joy of Your salvation and sustain me with a willing spirit.
v.13: Then I will teach transgressors Your ways,and sinners will be converted to You
AND there is more. Inspiring, don't you think?