There are other options other than the Priory for private psychiatrists. To be honest an NHS psychiatrist will probably change your meds as well. Going to A&E will involve hours of waiting to see the duty psychiatrist in difficult surroundings.
I think you should accept your symptoms as being generated by anxiety. I know they make you feel awful but reacting to them with fear and frustration will keep them going strong. Your nervous system is hyper stimulated and you are exhausted but nothing awful is going on physically. I do speak from experience and have been an inpatient. It's frightening and very isolating but you have to have faith that things will get better and you will not always live and feel like this.