If you can identify islands such as Barbuda and St Martin (which both saw the eye) on the map you will see that the width of hurricane force winds are wider on the north side of the track and equally the tropical storm force winds are wider north of the track.
If you continue this forward on the projected NHC track, it would seem unlikely Santo Domingo will get any tropical storm force sustained winds along with the bottom south west third of the country.
This is so because hurricanes consist of right and left semi-circles- The "Dangerous" and "Navigable Semicircles". The Dangerous Semicircle in the northern hemisphere is to the right of the storm's path. The leading portion of the dangerous semicircle is the Dangerous Quadrant.
Additionally, a hurricane's speed (speed of advance) is added to the wind speed (strength) e.g., a 100 mph hurricane moving at 15 mph would have a total effective windspeed of 115 mph in the "dangerous" quadrant.