I have personally been to South Goa during June - August and the beaches were deserted without any shacks or eateries. Visit North Goa anytime of the year, but visit South Goa only during between mid November and mid February.
Right time to visit south Goa is between June to October for it’s rainy season in Goa which gives you full view of lovely lush greenery around and best cool waters to touch. From south Goa it takes 3 hours to reach Gokarna. There are buses and cabs to drop you in Gokarna. If you are planning to take a cab the budget comes between 4K to 5k. If you are planning travel by bus the fare per head might come between 110rs to 150rs.