Delta has a pet line where you can call and get information directly from their specialized pet shipping department. I would call them and get information directly from them. Then you are getting good information instead of someone's best guess.
I've shipped dogs with Delta. They go in an airline approved dog crate and they get delivered to the cargo area, not the passenger area. At the other end, they get picked up at the cargo area.
When they are dropped off, the crate has to be inspected. I have to take the dog out. I don't know how you would do that with birds, except to maybe take the crate and the birds separately and place them into their shipping crate after it has been checked for explosives. I've put 2 crates mouth to mouth to move birds from one to the other and that worked well.
Also, I suggest that you phone your state veterinarian to see if your state has any requirements for bringing them in.