늘어지다 on its own is going slack or hanging limp. Combined with biting (물고) or grabbing (잡고, 붙잡고) it means clinging to something so the other side cannot get away — six lines here are 물고 늘어지다. Figuratively it is refusing to drop a subject: harping on one point, asking endless questions, going after one target and no one else. It is usually a criticism of the person doing it.