Yes you do pay for parking. I personally took a Lyft to and from which was fairly easy to do. You could also do the streetcar but it'll be a bit of a walk after you get off. Look on the New Orleans streetcar site for the closest stop from your starting location
Yes and parking is not cheap.
You can learn all about parking here: https://www.nationalww2museum.org/visit/plan-your-visit/parking
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