I picked 800 miles thinking you would be doing the inside which is much more convoluted.
Doing 700+ miles in 4 weeks breaks down to 175 per week or 25 miles per day every day. While I have done that it was a while ago and I tore up my shoulders doing it. You are obviously younger so it is probably within your capabilities but I think you might find it quite a challenge to keep up that average for a month with wind days and weather and rest days added in. Six weeks is much easier.
Finding a place to camp out on the oceanfront will be very very tough, particularly in North Carolina. Most of the oceanfront is developed or reserved as wildlife refuges. South Carolina is better. Georgia better yet - but past where you are going.
Going ocean you will near some surf experience. Do you have it?
Here is an alternative idea for you. Florida has a water trail all around the penninsula with lots of camping places on the west coast. Lots of information on where to stay along the way. Summer is hot a buggy but so is the Carolinas. Google it......