Big band is a category of jazz. Yes, jazz used to be rooted in those cities and the style of playing was different in each, and Kansas City as well. But that was in the 20s and 30s.
Dixieland jazz is the same as New Orleans style. The style of playing differently in each city does not exist now. Paul Whiteman and Ted Lewis were in the 1920s and the orchestration was different.
Broadway show tunes were not jazz. Here is one website to help you. The musicians who made distinct styles played various instruments, including sax, piano, clarinet, trumpet.
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