Praise be to Allah.
Yes, it is very necessary for you to enter Islam. This will solve all the problems, from making the marriage valid, through giving a united direction to the future upbringing of the children, to attaining success and happiness in the Hereafter. Your feelings of unease about taking this step may be due to the difficulty of leaving behind what you are used to and the religion that you have grown up in, or reluctance to go against your family and relatives, or fear of hostility and criticism from others, or concern about losing some worldly advantages. But all these issues will be easily overcome when you seek the help of Allaah and are determined to follow the truth. The wise person is prepared to make sacrifices and put up with difficulties for the sake of following the truth, because the truth is worth pursuing. Any difficulties encountered will become easy because the result is happiness in this world and the next, and Paradise the width of heaven and earth. Moreover, your marriage will help you to live in love and harmony with your husband (if he repents to Allaah for the forbidden relationship and becomes a religious person of good morals) and his Muslim family. There will be no dispute as to which religion the children will be raised in, and they will not feel that there is any conflict in the family, so they will be able to grow up free of the psychological complexes that result from the parents difference of religion. Other people have felt something similar to the feelings that you are experiencing now, as is reflected in the following story, which happened at the time of the Prophet of Islam (peace and blessings of Allaah be upon him). Anas reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said to a man, Become Muslim. He said, I feel that I dont want to. He said, Become Muslim, even if you feel that you dont really want to. (Reported by Imaam Ahmad, 11618; Saheeh al-Jaami, 974). This is the correct approach which people should have towards the true religion. For more information on the topic of marriage, please see Questions #3025 and #2527. We wish you every good thing and success. Peace be upon those who follow true guidance.