You're absolutely right - under the Old Covenant, cleansing - over and over and over again, was required, because the sacrifices that were offered only cleansed one until the next sin.
That was what the blood of bulls and goats accomplished.
The Blood of Christ, however, cleanses once for all, for all time, those who put their faith in Him.
The entire letter to the Hebrews is a treatise on the superiority of the New Covenant in Christ - His completely efficacious Work.
Repeated forgiveness and cleansing requires repeated sacrifices and the shedding of more blood.
That is not necessary in the New Covenant - the Blood of Christ was more than enough to provide forgiveness for the sins of the whole world. Every sin. For all time.
The Blood cleanses; the Spirit indwells.
Again, what must happen for blood to cleanse? It must be shed - in the Old Covenant, repeatedly. In the New Covenant ONE TIME.
If one says that there is a sin that must still be cleansed, one is saying that 1) they either need to offer an animal sacrifice, or 2) that Christ must come and shed His Blood again.
I know you know that neither of those are an option, and the letter to the Hebrews tells us why:
11 And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. 12 But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, 13 waiting from that time until his enemies should be made a footstool for his feet. 14 For by a single offering he has perfected for all time those who are being sanctified. (from Hebrews 10)
26 Such a high priest truly meets our need—one who is holy, blameless, pure, set apart from sinners, exalted above the heavens. 27 Unlike the other high priests, he does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself. (from Hebrews 7)
-JGIG