Does God still expect believers under grace to abstain from blood?
Yes. God still demands that even under grace, believers do not partake of blood. The life of the flesh is in the blood, and blood was given on the altar to make atonement for souls. Therefore, the requirement to abstain from blood continues for believers under grace.
Les explains that in Leviticus 17 God Himself speaks, declaring that the life of the flesh is in the blood and that blood was given on the altar to make atonement for souls. The blood of Old Testament sacrifices and Christ's blood had to be applied—sprinkled on the altar, the Ark of the Covenant, the Mercy Seat—and Christ presented His blood in heaven. Because of this, Les teaches, God still demands that even under grace believers do not partake of these things. The same context also warns against fornication and immorality, which God explicitly forbids in Leviticus 18.
For the life of the flesh is in the blood: and I have given it to you upon the altar to make an atonement for your souls: for it is the blood that maketh an atonement for the soul.