How old are your kids?
April will be likely too cold for local ocean or lake swimming so I'd focus on land activities your family enjoys, heated pools, hot springs, wildlife tours, mountain hikes, historic cities, museums, tennis, biking, golf if you are into that. Seems like a great time to visit anywhere has that nice flowers and landscaping.
Asheville could be anywhere from the 40s to the high 60s.
Cherry blossom peak prediction up in DC is April 11 or so, so somewhere gardeny down south might be good. Charleston, Savannah, SC/GA beaches? It's a very long drive to FL, even to Amelia Island, for a long weekend...
Book rooms now - we drove down on spring break Saturday and got shut out of rooms from Florence to the FL line, there's also a music festival in Savannah that runs to Apr. 4. We had to stay at a not nice motel near Ft. Stewart just to get some sleep on the way down.
http://www.savannahmusicfestival.org/