What do you call the man you are engaged to?

If I am reading your question right; the girl is officially his fiancee'; where he is her fiance'. Note the difference in the male/female spelling of the Words here.

In some cultures, being engaged is virtually the same as being married, and the man might then have some special way that he would address her directly. In the US this is not the case; engagement is not marriage and engagements can be called off without a lot of hassle. So if you are talking about how you would directly address her while engaged, you would use the same names or terms of endearment that you have used while dating.