That the earth is only 6 or 7,000 years old is not a given. I agree that the current creation is about that time scale. However, God's word implies that there was a creation prior to Genesis 1:2. It's known as Gap theory or the pre-Adamic creation. One proponent is Larry Ollison, I've watched his videos on the subject. Watchman Nee also spoke about this in His book, "The Mystery of Creation". It answers some difficult questions about the age of the earth and why it appears to be older than 7,000 years.
I do not believe in billions of years or even millions. However, the reality is that no one can prove the past because it is all circumstantial evidence. And it has no bearing on salvation. All I will say is that you can be born again and believe in a relatively old earth.