I would find a good person who does Clinical Hypnosis for flight phobia. Go for a few sessions, not just one, if you want it dealt with properly. As a Psychologist in the NHS, flight phobia often came-up when I was treating a patient for something else and I would devote 10 or 15 minutes of each of 3 or 4 sessions to addressing it with clinical hypnosis. It is a particularly easy issue to treat successfully that way. In the patients I treated, it never failed. I had one patient who was terrified of flying like you. She had only ever flown short haul. During each flight, she repeatedly vomited into a bag (hideous for those around her I imagine). Her husband laughed at her throughout this. If she needed the toilet, she was too afraid to unbuckle her seatbelt and leave her seat. She told me that her ‘lovely’ husband had booked a holiday in Canada for them and the children. She simply couldn’t conceive of going. I used hypnosis with her and in her final session before the holiday, told her that not only would she cope, but she would actually enjoy the flights. When she returned for her session after the holiday, she said “I enjoyed the flights, I really did”! The ‘moral’ of this story is, of course, that you can overcome this fear and even enjoy flying!
You won’t be able to be referred for this on the NHS for a number of reasons including of course, that it isn’t a mental health problem that effects your day-to-day life, but also because not all Psychologists (even if you did get to see one) know how to do hypnosis. When you see a hypnotherapist privately, a reputable one will not object to you having someone in the room with you if you like.