-(으)면서 runs a second activity alongside the sitting: sitting while nervous, while sweating, while turning something over in your head. In most of these lines the sitting is passive and a bit uncomfortable — 식은땀 줄줄 흘리면서 앉아 있어야죠, 'you just sit there dripping cold sweat'. One line uses it for taking turns instead: 번갈아 가면서 앉아야 하니까.