Can someone explain to me why the storm surge will be worse south of the eye?
It's because the forward velocity of the storm (how fast it's moving towards the coast) adds to the wind effects and storm surge.
Winds are onshore on the right hand side, off shore on the left hand side.
The winds south of the eye will be blowing onshore, piling up water along the coast. North of the eye, the winds will be blowing offshore.
South of the eye means wind will be coming from the southeast and then the East --- this will push the Water in that direction, Directly on to the shore.
The wind is going around the storm in a counter-clockwise motion. To the right of landfall, the winds are onshore and push the water up onto land, forming the storm surge. To the left of the landfall, the winds are offshore and push the water away. Once the storm moves further north, the winds swing around and cause a surge to the left of the landfall point, but that surge is not nearly as pronounced as the surge to the right of the landfall.
On the northern part of the eye-wall the winds will be blowing west - out to sea.
On the southern part of the eye-wall the winds will be blowing east,
pushing the water towards the coastline.