A big move is always a daunting task and which method of transport would be better will really boil down to your birds as individuals. We made a roughly 1200+ mile move with our amazon (newt and gecko) a few years ago. 3 solid driving days in a uhaul, but he did just fine and in fact seemed to enjoy the trip and spending all that time in close proximity to his people. When we stopped for the night, we'd let him out in the hotel room, which he rather enjoyed checking out new environments. He was also completely thrilled to be off his normal diet and to get to share in far more human food during the travel days than he normally would.
For our bird, while a flight may have been quicker, it undoubtedly would have been extremely stressful/traumatizing for him, much more so than the drive was. Not only would he have needed to be left with someone who could have dropped him off at the airport in the city we moved from so we could get to the new place ahead of his flight, but riding in the cargo hold of a plane, possibly with other animals of unknown health status, have his carrier handled by numerous strangers who may not be as gentle as us, exposure to the chilly outside air while being moved on/off the plane like a piece of luggage and across the tarmac to the terminal to be collected (and exposed to the fumes of jet fuel and god knows what else)... Nope! He was far better off with us, people he knows and trusts, and being cooped up in a carrier for a couple days.
It would be advisable if you do choose to drive to be sure your birds are acclimated to being in the car ahead of time. Our bird has loved the car since we got him, but I know some birds can be frightened, so a good way to prepare them would be starting as far in advance as possible, take them for daily short trips (drive around the block, as they become more used to it, drive around the neighborhood) to prepare them.
Good luck with your move and whichever route you go with.
