I fly with my Jardines all the time. Here's my advice:
1) This is the carrier I fly with.
https://www.celltei.com/products/in-cabin-airline-travel-bird-carrier-standard-size.html
I have tried other, cheaper carriers, and my Jardine's has chewed his way out of them. At an airport, my Jardine's actually crawled onto my shoulder because he chewed his way out of one. The celltei carrier is made with stainless steel mesh, no chewing out.
2) If you're flying in the US, you're limited to United Airlines or Delta. I prefer Delta. It's $125 per animal each way.
You are only allowed one carrier per person, and I highly recommend you have your Amazons in different carriers, so you will need two people. Delta doesn't require paperwork (they don't allow cockatoos), but you must call and reserve a space for your animal, as only 6 animals are allowed per flight, none in first class.
In addition, if you fly on a smaller plane, you might have less space for the carrier, though every plane I've gone on, the celltai has fit just fine.
3) Get an aisle seat. It allows you to swing the carrier under the seat easier.
4) TSA will require you to remove the bird so they can screen the carrier. Ask for a private room. Be warned, some airports have open topped private screening (atlanta) and I almost lost my Jardine's because of it. You will need to have your bird trained very well to get in and out of the carrier. Practice with your bird doing long card rides in the carrier first.
5) I don't put food or water in the carrier, too much risk of stuff getting knocked around. I put newspaper in the bottom, no toys. You can feed almonds / grapes by opening a side zipper.
6) Know the rules. Most flight personnel and tsa don't know their own rules regarding birds, so most of the time I explain their own policies to them, which helps (ie: hey, can I get a pet tag for my carrier, I need private screening to remove my bird, etc).
7) The celltai carrier has flaps you can close to keep it dark for the bird. should help if your bird starts to get noisy.
8) Be on the lookout for people with fake service dogs. It's better now that airlines have changed policy on emotion support animals, but I used to have so many incidents of dogs at airports trying to kill my bird through the carrier. Never had a problem with a legitimate service dog.
My Jardine's is a seasoned traveler and has flown more than most people have in their entire lives. Flying with a parrot is pretty fun and my guy enjoys it but there was definitely a lot of training and practice involved to make sure things worked out smoothly.