Can a Christian commit sin without being aware that their actions were sinful?
Yes it is possible for a person to sin without knowing it, including a Christian. Sin is any offense against God, and even when we lack a full knowledge of what offends God or the self-awareness to recognize that our actions are offensive, nevertheless we are still acting in a way that offends Him.
The Bible teaches this truth in both the Old and New Testament. In the Law given to Israel through Moses, the Lord described many requirements for the people of Israel. God told them not to murder and not to commit adultery, but Jesus said that hating a person was equivalent to murder in the sense that it was a sin coming from a similar sinful heart. Likewise, lusting was equivalent to adultery, Jesus said.
So even though Israel's Law did not prohibit lusting or hating specifically, those behaviors were still sin. Undoubtedly, many Jews were guilty of these unnamed sins and never knew it. Therefore, God specifically described unintentional sin in the Law:
Notice the Lord said that even when a person is unaware of their sin, he is still guilty. For this reason, God made a sacrifice available to the people of God in the Law to cover their unintentional sin.
Similarly, today Christians can sin in ways we don't fully appreciate, but our ignorance doesn't lessen our sin. We are supposed to remain forever in study of His word and under counsel of the Spirit so that we might better understand God's expectations and thereby cease sinning. Nevertheless, by our faith in Christ all our sins are forgiven - both those we know and those we don't know - so that we stand before Him eternally justified. Praise the Lord for His grace!