I have two questions:
- Did the Old Testament believers' experience being "born again" or "regeneration" as experienced with the New Testament believers through Jesus Christ?
- Are born again believers still considered to be "sinners" or "saints" and why?
There are several passages which use the allegory of birth to explain the change that occurs to person who follows Christ. Jesus stated, "Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God" (John 3:3). Here then is our first clue. Seeing the kingdom of God is dependent on being born again. A bit later, Jesus makes the statement more firm when he said, "Most assuredly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God" (John 3:5). Now we know what constitutes being born again: being born of water and the Spirit. We also know that you must be born again to enter the kingdom of God.
This leads to a side question: What is the kingdom of God? The writer of Hebrews tells Christians, "Therefore, since we are receiving a kingdom which cannot be shaken, let us have grace, by which we may serve God acceptably with reverence and godly fear" (Hebrews 12:28). Reading backwards, we find from the context that this kingdom is another name for the church. "But you have come to Mount Zion and to the city of the living God, the heavenly Jerusalem, to an innumerable company of angels, to the general assembly and church of the firstborn who are registered in heaven, to God the Judge of all, to the spirits of just men made perfect" (Hebrews 12:22-23). This imagery of the church is one where Christ reigns as king and Christians live as citizens. "Giving thanks to the Father who has qualified us to be partakers of the inheritance of the saints in the light. He has delivered us from the power of darkness and conveyed us into the kingdom of the Son of His love, in whom we have redemption through His blood, the forgiveness of sins" (Colossians 1:12-14).
Just as Israel was God's people and to be a part of that nation one had to be physically born into it, so to is the church seen as God's people under the new covenant. "But you are a chosen generation, a royal priesthood, a holy nation, His own special people, that you may proclaim the praises of Him who called you out of darkness into His marvelous light; who once were not a people but are now the people of God, who had not obtained mercy but now have obtained mercy" (I Peter 2:9-10). Like the Israelite, a Christian must be born a citizen, but instead of a physical birth, a Christian must be born spiritually.
To be an Israelite, one merely was born to Israelite parents. What it means to be an Israelite was then taught to the child as he grew up. To be a Christian, one must be born a second time, this time spiritually. He is taught what is needed to be a Christian before he makes the commitment. "No more shall every man teach his neighbor, and every man his brother, saying, 'Know the LORD,' for they all shall know Me, from the least of them to the greatest of them, says the LORD. For I will forgive their iniquity, and their sin I will remember no more" (Jeremiah 31:34). Therefore, no, the Israelites did not have to be born again to be under the law of Moses.
For your second question, see: You Must Be Born Again. To be "born again" is to be saved from your sins.