Praise be to Allaah.
The marriage contract is valid and it is no need from an Islamic point of view to renew it simply because of a doubt. But what is mentioned in the first part of the question, about kissing the woman during the engagement period – if this was before the marriage contract (‘aqd or nikaah), then it is haraam. The same applies to being stimulated manually by the woman. But if it was after the marriage contract then there is nothing wrong with kissing. With regard to adulterers marrying one another, there is nothing wrong with that, after waiting out an ‘iddah period and after both parties have repented. Allaah says (interpretation of the meaning): “Women impure are for men impure… and women of purity are for men of purity…” [al-Noor 24:26 - Yoosuf ‘Ali’ translation of the meaning]. Repentance is essential on the part of both, but it is not permissible to conclude the marriage contract until after the waiting-period and making sure that the woman is not pregnant as a result of zinaa. Once that has been established, there is no reason why they should not marry one another.
In your case, as you describe it in the question, there is no need to repeat the marriage contract, but you must both repent to Allaah for the forbidden relationship which you engaged in before you were married. And Allaah knows best.