All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Your cousin is marriageable to you and it is permissible for you to marry her. Feeling that she is like your sister does not make her prohibited for you. So, if she is religiously committed and of a good moral character and you can marry her, then this will be good for you and will achieve the wish of her parents and yours; for your words imply that your parents too want you to marry her. Also, Allaah The Almighty may bless that marriage for you. As for what you feel, it may vanish after marrying her by the permission of Allaah The Almighty.
If you predominantly think that there will be no love and affection between you and her after marriage, then it is better to cancel your engagement to her and try to please your parents and hers regarding that. There is no harm in canceling the engagement if there is a need for that. For more benefit, please refer to Fatwa 82215.
As for the Christian woman, we advise you to beware of committing something Allaah The Almighty prohibited with her. Also, speaking to her is not permissible unless there is a need and according to that need. If you go beyond that, this may lead to temptation and thus beware of that. Her conversion to Islam is something good and we hope it comes true. However, we advise you to entrust this matter to some trustworthy women to tell her about the truth of Islam and encourage her to enter Islam. You should not link her conversion to Islam with marrying her. For more benefit, kindly refer to Fatwa 105117.
Anyway, if that Christian woman converts to Islam and you would like to marry her after embracing Islam, then try to convince your parents to agree to that and do your best in that effect. If they agree, then all praise be to Allaah, but if they insist on rejecting and they have a legitimate excuse, then you have to obey them, otherwise it is permissible for you to marry her, even without their consent. Nevertheless, if you prefer to comply with their opinion, then it is really more appropriate and Allaah may grant you guidance and success due to that and He may grant you a pious woman who will make you happy. For more benefit, please refer to Fatwa 87553.
Allaah Knows best.