I think you misunderstand the question. Jesus taught and so do the apostles that repentance and belief are all that is required to receive salvation. It is the gift of God not of works lest anyone would boast. We cannot earn salvation. In that you are correct.
But we can choose to live by the flesh even though we have the spirit. Paul talks about this at great length in Romans. That we have the flesh and the spirit in us at war. But the good news is we are no longer a slave to the flesh. We do not have to walk in the flesh. Everyday and every minute we have a choice that the unbeliever does not. But we can by our own choices to live after the flesh and not the spirit we have received. So no we cannot wilfully engage in walking after the flesh once we have received the spirit because we bring ourselves into enslavement again into sin and the flesh.
Therefore, there is now no condemnation for those who are in Christ Jesus,because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death.For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh,in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit.Those who live according to the flesh have their minds set on what the flesh desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires.The mind governed by the flesh is death, but the mind governed by the Spirit is life and peace.The mind governed by the flesh is hostile to God; it does not submit to God’s law, nor can it do so.Those who are in the realm of the flesh cannot please God.You, however, are not in the realm of the flesh but are in the realm of the Spirit, if indeed the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, they do not belong to Christ.But if Christ is in you, then even though your body is subject to death because of sin, the Spirit gives life because of righteousness.And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies because of his Spirit who lives in you.Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it.For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdhaeeds of the body, you will live.For those who are led by the Spirit of God are the children of God. - Romans 8:1-14
Bible Gateway passage: Romans 8:1-14 - New International Version
So we have an obligation to live after the spirit. Note that it is OUR OBLIGATION. It's not automatic. If you live after the flesh you will die. But those who are led by the spirit are the children of God. So our obligation is to live after the spirit and not after the flesh. This all happens AFTER belief and not before. Because before belief it is impossible to live after the spirit.