No, the position is that sacrifices are welcome, and a part of a relationship with God. Psalm 50 (51 in your bible) is a psalm about repentance. Just like it is echoed time and again in the prophets, sacrifice isn't a path to forgiveness, it is part of the relationship that happens after you repent. You cant live a life of sin and blood, then offer sacrifices and hey presto all good. This psalm is the same message, once David repents, then he can bring his sacrifice to the altar. This is the same thing taught by the Lord - not, dont offer sacrifices, but "first be reconciled to your brother, and then come and offer your gift."
You can't say "God wants you to repent" and ignore the "and then offer the sacrifice."
As for the offering, St Paul tells us our offering is of ourselves - our rational worship, the worship of rational beings. Accordingly in the Liturgy we say multiple times that we offer ourselves, and each other, and our whole lives unto Christ Our God. The bread and wine are brought and we pray for the Holy Spirit to come down upon us and upon the Gifts, the Gifts are blessed and made to be the Body and Blood of God, and offered to Him as a sacrifice along with our sacrifice of praise. He is the One Who both offers and is offered, the One Who is received and is distributed. So we give ourselves to Him, He gives Himself to us.
Remember that the whole sacrificial system, the Tabernacle, was a copy of what Moses saw in Heaven, on the Mountain. In the Liturgy we participate in that, which is specifically the worship of the Heavenly Tabernacle, where Christ is the offering.
