Coorg will be more of mist, fog and lush greenery in October. Weather will be cold. It’s one place where you get to see the peaks in this month, landscapes and great valleys. Gokarna is full of cool air, great beach views, ideal beach trek time, and to indulge into water sports. One such ideal time to visit Gokarna. From Coorg Gokarna is 7 hours away. My advice would be to club both, else if you are planning to visit only one place I would suggest Coorg.
Both are great places. It all depends on your travel style and personal preference - mountains or beaches. But considering the time and weather, I would prefer Coorg in October and Gokarna in Nov-Jan.