ACA maps will give you all that info. You can go to google maps to look up campgrounds in the cities you will traverse. They have listed phone numbers.
I have cycled the Atlantic coast three times. I did not stay in campgrounds. I free-camped wherever I could. To my way of looking at money matters, what they are asking for a bare hard piece of terra firma for one night in a tent is crazy, way too much. I would be very surprised to see any private c'ground on the east coast offering hiker-biker rates like parks on the pacific coast. The Pacific coast bicycle route is a lot more scenic and interesting.
Hostels, cheap public park camping, room, beaches, hills, great scenery, fresh air.. In 1990 I left southeast coastal Florida on May 1st. I was in Bangor, Maine on May 22nd. It was still freezing a couple of nights in the NE part of the route. April would be cold, but perhaps not prohibitively cold. Bundle up and you should be ok.