I'm only going to answer for Arkansas; my knowledge of camping rivers in the other states is too weak.
As Kim suggested, the Buffalo would be an excellent choice, but maybe just a bit easier than you were asking for. BUT -
The Buffalo is the only river in Arkansas where you could mount a 5 day expedition; all of our other streams have dams, join a major river like the Arkansas (which has a lock and dam system, no WW) or just go out into the flatlands with no WW after a couple or at most 3 days.
The Buffalo has great scenery and and the translucent blue-green water that was typical of all our Ozark streams in years past (except after rains).
The Buffalo runs its entire length through public land (National River, which means it is part of the US National Parks system), so you'll never have to worry about accidentally camping on somebody's property.
The Buffalo is not "developed", you'll see only an occassional road crossing, and pass a few campgrounds.
You can camp anywhere that suits you; most like to stay on the large gravel bars. Choose one that backs up to a hill so that you have an "escape route" if it rains. While rare, the river is capable of rising 40' in a few hours. Unless the sky is clear blue and you are 100% sure it is not going to rain anywhere within 100 miles, consider carrying your canoe up the gravel bar to a point higher than your tent. (This insures you will be awake BEFORE the canoe floats downstream.)
Watch this thread; our resident "Buffalo river camping expert" Richard will weigh in; I bet if I mention it he will post the lastest version of his spreadsheet summarizing mileages and with links to gages. (But take some of his advice with a grain of salt, unless you think camping should be done with the same level of comfort you expect at a Hilton.)
Buy a book; there are a number of side hikes worth doing, depending on what stretch of river you are on. Also, take time to walk up to the old country store at Gilbert; on a 5 day trip you will almost certainly go by there at some point in your trip. You should also take the time to scramble up the rocks at the "Nars" (the Narrows"). It looks like just another bluff line from the river, but when you get to the top, you'll be glad you did. Hummm - do you see the pattern emerging - "take the time to...". That is the beauty of the Buffalo, taking the time to just immerse yourself in it. Also on the "must hike" list - Hemmed in Hollow, if you launch at Steel Creek. While still in your car, go to Lost Valley and hike the trail there up to the waterfall; take flashlights and go into the cave and make your way back to an underground waterfall (no hard-core caving experience is needed).
Water levels are rain-dependent, but March should provide a good window. Don't get "locked in" on exactly which stretch you will do, stay flexible so that you can make your final decision when you arrive and see what the level is and what rain is in the forecast. If there is enough water, you should start at Steel Creek; but if there is not, you can start lower down, and continue your trip all the way to the White River.
You might also want to look into the Current River, Eleven Point, or the Jacks Fork, in southern Missouri. In summer, they become pretty "wild" on weekends, but in March they might provide a good opportunity for you. There would also be some options like, 3 days on the Buffalo, 3 days on the Current, etc, which could come into play if you are willing to consider two shorter expeditions instead of one long one.
For my money, I don't think you could beat a 5 day trip on the Buffalo.