Is there such a thing in our current era as "milk al-yameen" (rightful possession) such that a man becomes lawful to a woman merely by her saying, "I have given you my soul as a free Muslim woman, belonging to no man," and he says, "I accept," and does this permit conversation between them as spouses?

Religion prohibits relationships between sexes outside of marriage and forbids dating programs to prevent discord. The questioner erred by establishing a dating relationship with a young man unrelated to her. Furthermore, her statement "I have given you my soul" is incorrect, as it is not permissible for a man to marry a woman without her guardian's permission, nor is it permissible for a woman to offer herself to a man; this was exclusive to the Prophet, peace and blessings be upon him. Her statement "right hand possess" is also incorrect, as a free Muslim woman cannot become an enslaved handmaid. The reason for enslavement is disbelief and fighting against Allah and His Messenger. A marriage not founded on legitimate Islamic principles is doomed to fail. The questioner must repent and distance herself from this young man.