I am from Badinan, but Badinan is only a small part of southern Kurdistan. It only consists of Amediye, Duhok, Akre, Zaxo, Barzan and other villages in between. There’s still Hewler, Kerkuk, Slemani, and others that make up southern Kurdistan =)
But yes, most cities in the Badinan are really close to the border with northern Kurdistan. My city in particular is really close to both northern and western Kurdistan (Amediye).