We also relocated cross country with 2 cats last year, one being a Maine Coon. We flew with Delta. Make sure when you book your flight, the plane is big enough where he/she can fit under the seat. We had to rebook our flight because our original plane was too small.

We bought this carrier: Sherpa Deluxe Pet Carriers

Make sure you get the medium size. We originally bought the small but it was too small for our Maine Coon.

We also took our cats to the vet before moving. I think you need to take them to the vet and get them cleared for flying, might be an airline regulation. We asked about sedating at the appointment. Sometimes it could give them more anxiety, so our vet gave us a little bit of sedation medication to give them to see how they do on it. They also gave us some pheromone wipes to wipe the carrier with, it supposedly calms them down. You might want to try that.

One of my cats peed so you might want to get pee pads like someone suggested.

Good luck!