First of all, Tab, many of us have been where you've been. I lost both of my parents a few years back after their very long and wonderful lives ended. It's a tough time to get through and I thoroughly understand the need for a road trip to get your mind re-set, so.....
I second what Lane said: Go diagonally SE from Ohio to end up in New Bern, NC, then out to Moorhead City and the lower outer banks. Take RT58 south on the outer strand down to Cape Carteret, RT 24 up to Jacksonville, NC, to the Southern Coastal Route, RT 17 south. (East Coast route 1 from Maine to Key West goes inland just North of here and returns to the coast in Jacksonville, Florida, so the coastal route you want is RT 17 all the way to Brunswick, GA.)
Stay on RT 17 South, just skip right through Myrtle Beach (It's kind-of Honky-Tonk and really not typical of "The South") and head to three of the four best towns in the South:
Charleston, SC: Spend a day or so there - Lane knows the good restaurants - Take the "Trolley Tour" of the city (it's a big college town), but then wander (on foot) through the old section south of Broad Street (called "The Battery") - look through the gates into the back yards of the old homes...The gardens are spectacular...The homes are, too. There are many churches in Charleston (The "Holy City") - they're all beautiful and most of the old, pre-Civil War buildings are still there. All of those old buildings were built by the extraordinary skills of slaves.
Beaufort, SC: Take RT 21 south from RT 17 at Garden Corner to Beaufort. Much like Charleston, but in miniature - Take the Horse-drawn carriage tour through the old section - pick it up in the parking lot of waterfront park. Ask on the tour about a man named Robert Smalls. Stay overnight at the Best Western downtown, right on Waterfront Park or at the Rhett House Inn B&B (yes, the same Rhett family as in "Gone with the Wind"). Look around and you'll see scenes from "The Great Santini", "The Big Chill" and "Forest Gump", all filmed there. Visit the National Cemetery and find Donald Conroy - The real "Great Santini" and the father of Pat Conroy, the author. Eat at the Breakwater or Low Country Produce Cafe on Carteret St. or Brick's Grill on Boundary, St. and, if it's a Friday evening, stop by the Beaufort Classic Car Club "Cave" at the corner of Middleton and Hay streets. Ask for Rob Hilton - Tell him I sent you. First Friday of the month they move to the Grayco parking lot on Lady's Island for a car show starting at 5pm. A great side trip is to visit the Penn Center out on Lady's Island, where Martin Luther King wrote his "I have a Dream" speech. Stop at "Gullah Grub" nearby for lunch and get the fish chowder - it's legendary. If it's close to Saturday night, go there for their Fish Fry.
Savannah, GA. Later, take RT 170 south out of Beaufort to Savannah. The "Historic District" is 6 blocks by 8 blocks and every other block is a public park - they are all stunning in the spring when the Azaleas are in bloom and not bad in the Summer with the roses. Take the trolley tour of the city and find out why General Sherman did not destroy Savannah on his famous "March to the Sea" (He spared Charleston, too, but for a much different reason). Pick up the tourist trolley at the Visitor Center on MLK BLVD. See Forsyth Park (where Forest Gump sat with his box o' chocolates) but DO NOT eat at Paula Dean's place. 
Instead, go to Mrs. Wilkes Dining Room for real southern cooking. Find Johnny Mercer's house - It was used in the film "Midnight in the Garden of Good and Evil". Ask about Valerie Fennel Aiken Boles or "Minerva" in the film (she came from Beaufort). Stay at a B&B in the Historic District.
Go south from Savannah on I-95 just a little bit to South Newport, then pick up RT 17 again all the way down to Brunswick, GA. Lots of beautiful little coastal fishing villages all along the way (lots of shrimpers) and that area is unique to the South. Lots of fish shacks for food and they're all great (Skipper's Fish Camp in Darien). When you get to Brunswick you've reached the end of the Scenic SE coastal route and it becomes scenic only on A1A down to Key West and quite repetitive.
So from Brunswick, if you have the time, you could keep going down the coast a while or, if it were me, go back to Savannah and go back the way you came (it always looks different going in the other direction) or pick up I-16 to Atlanta and then up to Ohio, or I-95 back to Charleston, I-26 to Charlotte and up to Ohio Either way it's 2 - 3 days back to Ohio.
Have a wonderful trip! Gordon
