Is it permissible to marry a man who is already married and has two children, and who has divorced his wife twice verbally only, and she is still living with him, and he justifies this by saying that there is no one to care for their two children, with his promise to send her to her family immediately after his marriage [to me]? And would I be considered sinful by agreeing to this marriage?

The Islamic standard for choosing spouses is piety (din) and good character (khuluq). It does not matter if he has been previously married or divorced, based on the saying of the Messenger of Allah, peace and blessings be upon him: "If there comes to you one whose character and religious commitment you are pleased with, then marry [your daughter] to him. If you do not do so, there will be tribulation on earth and widespread corruption." It is necessary to inquire about the suitor from trustworthy people who know him. If they praise him well, then there is no harm in agreeing, and there is no injustice to his wife in that. Seeking guidance (istikharah) is recommended. If the marriage took place without a guardian (wali), there is a difference of opinion on this matter, and the majority of jurists stipulate the guardian for the validity of the marriage. A woman is not divorced merely by her desire for divorce.