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That sounds very uncomfortable OP. Where will you sleep? Will it be in summer, winter, in between? If summer you will roast in AZ sun, if winter NY cold will get you. Very inconvenient way to travel my man. I'm a walker and a runner erryday, but I'd never try to tackle that sort of distance on foot. It sounds foolhardy and risky.
That said, if you must do it, maybe try to find a very small 1 person tent and keep your light low at nights when you get caught outdoors. Police will probably nab your ass for vagrancy in a number of states if you just camp by the side of the road. Not to mention people will pull over and probably assault/rob you for shits and giggles.
Best of luck tho.