Having been a long time resident of Southport, I can tell you that it is VERY dog friendly, especially in the downtown area. The downtown area has sidewalks for walking. Also, you can take your dog to a little section of beach on the Cape Fear River right in the area known as "The Yacht Basin." I used to do that every day with my dog. And while you are there, be sure to look for beach glass and pieces of old broken pottery that has washed ashore from sunken ships. I am serious! It's like looking for a needle in a haystack but when you find a tiny piece of a fine bone china plate, it is really exciting.
If you have any questions about what it is like to live in Southport, do not hesitate to call me.