My grandfather Lorenzo came to New York around 1900 and his brother remained in Corleone. The family lived in Little Sicily on 107th St. Manhattan, my father brought us to San Diego, CA in the early 1960's.
It is rare to to find a Badami whose roots are not from Corleone. I am surprised your cousin was rebuffed by the family. Perhaps she inadvertently insulted or slighted them in some way. Does she speak Sicilian? They do not speak English and find it odd when Sicilians don't speak the language. However, I have traveled there three times, discovered distant cousins I did not know existed and was welcomed with open arms and my lack of good Sicilian was not a problem. In fact, they refused to let us and our friends stay in a hotel and and invited us to stay in their home.
Did your cousin make contact with Franco Badami who owns the clothing store on Via Don G. Colletto? I suggest you send him a letter in Sicilian.
By the way, did you do any Ellis Island research at ellisislandrecords.com?