To get over airport anxiety be sure to get to the airport early and allow yourself enough time to relax at the airport.
Arriving late or close to the time the airplane leaves can increase your anxiety and so try to arrive to the airport at least 1 hour to 2 hours before the airplane leaves.
This will allow you time to get through the check in and security and allow you time to relax for awhile before you're scheduled to board the airplane.
Then to also help yourself get over airport anxiety take some time to find a seat, put your phone away, close your eyes and breathe.
And by breathing deeply you can slow down your heart rate, relax your body and ease your body out of anxiety mode.
It can be stressful and scary at first if it's your first or second airplane ride but after the first few airplane rides and first few times going through security at the airport etc it gets a little easier for most people.
I used to get a lot of anxiety and was really scared when I first flew but after flying a few times without any issues I become less scared and I don't have as much anxiety as I used to.
You could also take some sleeping pills when on the airplane to help you sleep.