My sympathies on the death of your husband.
Regarding adultery, this is committed when there are sexual relations between two people, one or both of whom are married to someone else. You would have to know the other person was married for you to be guilty of adultery. However, if there are sexual relations between two people who are assumed by each not to be married, and they aren't married to each other, then the sin of fornication is involved. If there were no sexual relations, then neither of those sins applies. You shouldn't feel the need to say here whether there was sex involved or not. Let's all assume there wasn't.
Regarding dating this man...in my opinion, dating is meant to determine suitability for marriage. If you are Catholic, you are required to be married in the Catholic Church, or receive a dispensation for form, or the marriage will be invalid in the consideration of the Church and you will be considered to be committing adultery. If the man's prior marriage has not been examined by the Church for validity and has not been declared null, then you do not have the possibility of marrying this man in the Church. Therefore, dating him is a problem, since there is no place this relationship can develop that will legitimize anything but a non-physical relationship.
Of course, if all you are doing is maintaining a platonic relationship and you see no danger that something else will develop, then it is not necessarily a problem.
Most people's sense of history goes back to breakfast time - Benjamin Netanyahu