Regeneration
Other than baptism, is there another means of regeneration? What exactly is meant by regeneration? If after the fall into sin man is incapable of accepting the Law or understanding what the Word of God proclaims because he in fact hates God, how can he ever accept God's grace? It is my understanding that we do not in fact accept God's grace, but it is a gift given to us that is ours only to reject. So then, how can we be willing to be baptized if we by our natural state reject God and what he stands for?
“Regeneration” describes Christian conversion, brought about by the Holy Spirit, as “rebirth.” People are naturally spiritually dead (Ephesians 2:5). When the Holy Spirit changes hearts and gives people saving faith, spiritually dead people become alive spiritually. The Bible describes that change as “rebirth” or being “born again” (John 3:5; Titus 3:5; 1 Peter 1:23).
The Holy Spirit brings about rebirth as he works through the word of God alone or the word of God attached to water (Baptism).
In regard to your last question, when the Holy Spirit brings an adult to saving faith through the word of God alone, there would be good reason for that person to want to be baptized: to do what Jesus says and to receive an additional guarantee of God’s forgiving love. That desire to be baptized comes from a person whom the Holy Spirit has already brought to faith through the word of God alone.
In all of this, we want to keep in mind that Christian faith is a gracious gift of God (Ephesians 2:8). Thanks be to God for this gift.