It's doable, provided you stay out of the Gulf Stream
. You'd need to hug the coast. The Stream runs north at 2-4 knots; you don't want to be bucking it's current
. And, you particularly don't want to be in the Stream with a north wind
(which would be favorable to you, but would cause short steep seas which are very, very uncomfortable.
The Gulf Stream
generally swings in pretty close to Cape Hatteras shoals, too, so you'd have to be in real close to avoid it. Much closer than would be safe, IMO.
In your boat, if you're not in a particular rush I'd favor the ICW
where you can take your time, see very interesting places, and be near protection if a storm blows up.
And, don't forget, late June is into the hurricane
season. So, you'd have to watch the weather
Again, it's doable, but wouldn't be my first choice -- particularly not in a 28-footer :-)