I think the form Adamus, Adami is a later one and doesn't occur in the Bible itself or in authors of that time. In the Bible, you find indeclinable Adam and Adam, Adae.However some names that don't fit as well seem to be more variable. For instance, I've seen "Adam" (as in Adam and Eve) treated both as indeclined (nominative Adam, genitive Adam, etc.) or as Adamus, Adami. It really depends on the specific name.
From what I've seen in Medieval and later Latin, the feeling I've got is that most names are Latinized as second declension (or first for female names) unless they already have a great likeness to third declension.