The Greek word here is συνείδησις (suneidēsis), which can mean both "conscience" and "consciousness."1 The word also appears in Hebrews 10:2, in addition to the other verses you cite, but here the ESV translates the word as "consciousness" rather than "conscience":
ἐπεὶ οὐκ ἂν ἐπαύσαντο προσφερόμεναι, διὰ τὸ μηδεμίαν ἔχειν ἔτι συνείδησιν ἁμαρτιῶν τοὺς λατρεύοντας, ἅπαξ κεκαθαρμένους;
Otherwise, would they not have ceased to be offered, since the worshipers, having once been cleansed, would no longer have any consciousness of sins?
As verse 7 states, the high priest would only atone for the unintentional sins (ἀγνόημα - agnomēma) of the people:
Into the second only the high priest goes, and he but once a year, and not without taking blood, which he offers for himself and for the
unintentional sins of the people.
According to the Law, however, intentional sins could not be atoned for by ritual sacrifice:
If one person sins unintentionally, he shall offer a female goat a year old for a sin offering. And the priest shall make atonement
before the LORD for the person who makes a mistake, when he sins
unintentionally, to make atonement for him, and he shall be forgiven.
You shall have one law for him who does anything unintentionally, for
him who is native among the people of Israel and for the stranger who
sojourns among them. But the person who does anything with a high
hand, whether he is native or a sojourner, reviles the LORD, and that
person shall be cut off from among his people. Because he has despised
the word of the LORD and has broken his commandment, that person shall
be utterly cut off; his iniquity shall be on him (Numbers 15:27-31)
Verses 9-10 reiterate this. The ESV here reads:
According to this arrangement, gifts and sacrifices are offered that cannot perfect the conscience of the worshiper, but deal only with food and drink and various washings, regulations for the body imposed until the time of reformation.
But I think the translation may slightly miss the mark. The Greek text does not say "perfect the conscience", but rather "perfect the worshipper" (ελειῶσαι τὸν λατρεύοντα); and while "reformation" (probably borrowed from the KJV) may resonate with some readers, it is probably better rendered as something like "make right" (ὀρθός - orthos, meaning "straight" or "right" - is the root of the verb διόρθωσις). A more literal translation of this passage might be:
καθʼ ὃν δῶρά τε καὶ θυσίαι προσφέρονται μὴ δυνάμεναι κατὰ συνείδησιν
τελειῶσαι τὸν λατρεύοντα, μόνον ἐπὶ βρώμασι καὶ πόμασι καὶ διαφόροις
βαπτισμοῖς καὶ δικαιώμασι σαρκός, μέχρι καιροῦ διορθώσεως ἐπικείμενα.
... According to which both gifts and sacrifices are being offered,
which is not able to perfect the [one] who worshippeth, as
concerning conscience, only in food and drinks, and diverse ablutions
and ordinances of [the] flesh being imposed [on them], until the
time of setting things straight.2
So we conclude from the text itself that the contrast is between a time in the past when things were not "straight" or "right" (ὀρθός) and the time of "straitening" (διόρθωσις). Before that time, consciousness of sin could not be erased by the atonement of the high priests of old (cf. Numbers 15:27ff above), but was erased by Christ who came as High Priest (Hebrews 9:11)
This line of reasoning is explained in one modern Orthodox commentary:
Perhaps the use of "sins of ignorance" [ἀγνόημα] points up to the fact that deliberate sins were not forgiveable (Numbers 15:27-31), so as to make a contrast with the effectiveness of the sacrifice of Christ, which would cover all of man's sins: "... the blood of Jesus Christ His Son cleanseth us from all sin" (I John 1:7).
1. See, e.g., Bauer's Greek-English Lexicon of the New Testament and Other Early Christian Literature
2. Orthodox New Testament translation (not to be confused with Orthodox Study Bible)