Gambit,
I believe whatever the Bible says, and if I find something in the Holy Scriptures that I did not know, I am very willing and happy to learn, and I will change my belief instantly when I find something I did not know, or had a wrong idea about.
To hold onto long believed doctrines that are proved wrong, is believing in The Sacred Cow, or a Home Truth. These doctrines can be pseudo doctrine, or even worse, Cacodoxy.
The belief in reincarnation, I believe is belief in a Cacodoxy, because it goes against everything the Bible says about the life that God has given us. Reincarnation is also called Migration of souls from one person to another, or even to animals.
To begin; neither mankind or animals HAVE a soul. The Bible states very clearly at Genesis 2:7, that God formed Adam from the dust of the earth, then blew into his nostrils the breath of life, and the man BECAME living soul. The soul is the life of the person, animals are souls, Gen 1:20. God the Creator even has recorded that His life is a soul, Ps 11:5, Pr 6:16.
When explaining how the soul is formed, Traducianism is closest. The soul comes from the parents, who have been given the ability to procreate, by Jehovah God.
There is no need for souls to migrate, for God has promised that there is going to be a resurrection of dead ones, John 5:28,29, even a resurrection of both the righteous and the unrighteous, Acts 24:15. The term resurrection means a standing up again to life. Only the ones that God classes as evil will not be brought back to life.
There is one scripture that bears directly on the idea of reincarnation, Rom 9:10,11. Consider what verse 11 tells us; it says that before Jacob and Esau were born to Isaac they had done nothing, either good or bad. This means that they did not live before as souls, because if they existed before, they would have had to do something, something either good or bad. They or their souls never existed before they were born to Isaac and Rebekah.