New Orleans can be done without a car. The streetcar track runs parallel to the the river and lots of things are walkable from it, there are buses in the centre and taxis also abound. It's the only southern city I know with any pretense at public transport. But I don't think you can go out to see the 'real' Louisiana without a car.
Atlanta has, I think, a 'metro/tram' system left over from the Olympics back in ... But historically, culturally, Charleston, Savannah and even Richmond are much richer...
Philadelphia might be ok without a car but you'd have to transit there from somewhere else (by train?) or fly in directly.
San Francisco was ok without a car also. But what about Dallas and Austin - public transport just cannot cover that breadth and there are no 'trains'. Also Miami or Denver - destinations but without any serious public connections ... You could go to Vegas but you'd be 'trapped' there.
I think the sprawl of LA might really need a car to see all the highlights: the fancy houses, the recording studios' front doors ... (Lincoln Lawyer... he realises how much time he's wasting driving back and forth to his office)
Plus, how do you visit the vinyards? Sonoma? Are there buses laid on for that? What if you want to go down to Tijuana?
As others mention, you are not visiting the states but rather the bigger cities. The cities are in such different areas that you are having a taste of the different regions but true Americana is in the small towns and there is not really any non-car connection between them.