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Yes, the Gulf of Mexico is nearby, but you are talking about the Mississippi Delta here, with the zillions of gallons of silt it deposits in the Gulf as it enters. That's why the Gulf coast closer to New Orleans isn't particularly beach-y.
Definitely just do New Orleans and save the beach for another trip unless you have a lot of time. We spent a week in NOLA and still didn't see everything we wanted to.
Just muck from NO to the Gulf. No beach. Even if you could reach it, you wouldn't swim in it. We lived there a decade and when we went to beach, it was to Destin. And don't even think about stopping on the MS Gulf coast. UGH!