Your theory makes a lot of sense. I had always thought the further back one goes one would find a core of Adami ancestry in Italy [probably northern and central areas] and/or South Eastern France. I had heard the theory from other relatives that originally Adami's were from the southern Balkans.
The migration of peoples throughout Europe has a long and fascinating cause and result. You will find that almost all families were affected by this movement. Hence one gets new branches of families developing in more unusual areas. More often than not the movements were forced. Just go back to the 1800's and you will find that many people rarely moved from the areas in which they were born. So when we talk of the mass migrations [forced or otherwise] in Europe in the period 500 AD to 1500AD we are talking of very uncommon displacement of people. People only tended to move when they had to, not however, for marriage, which is rather odd. I would class the crusades as a mass migration of peoples.
I think all Adami's are related in some way. You just have to go back far enough with your research.
Another point I have thought of often is when did Europeans in general adopt surnames? In Britain I know it is generally said that surnames came from things like occupations. Adami - means what? or came from where?