HPA axis dsyfunction isn't a disorder or syndrome. It quite simply doesn't happen. Certain endocrine disorders obviously effect the HPA axis - adrenal insuffiency for example (not adrenal fatigue which is psudoscientific rubbish).
Abnormal functioning of the HPA axis has been implicated in certain affective disorders (Generalized anxiety, Severe and/or Psychotic Major depression).
There is also Dysutonomia which can be caused by various disease processes and can also be direct pathology of the autonomic NS. While not often having anything to do with the HPA system it often produces certain similar symptoms. Most of these disorders are very rare.
If you do manage to convince a doctor you are suffering from HPA axis syndrome they should be imediately stopped from practicing medicine.
It is unlikely you have anything other than a psychiatric disorder. GAD doesn't require there to actually be any psychic anxiety. A somatiziation disorder isn't out of the question.
A Neuropsychiatrist would be absoutely no help. They specailize in psychiatric sqeulay from Neurological disorders, and the diagnosis and management of Neuropsychiatric disorder (traumatic brain injury, Sezuire disorders, Somatiform disorders, physiological sleep disorders, parkinsons, Dementia syndromes etc...). Occationally Neuropsychiatrists may become involved in other disorders causing psychiatric symptoms (that aren't psychiatric in orgin) but this tends to be delt with by General psychiatrists.
You absoutely should chase down any REASONABLY possible non-psychiatric disorders. Standard bloods should be taken. Thyroid function tests (hypothyroidism is the most common disorder causing psych symptoms), Liver function tests, Full/complete blood count, fasting glucose/HBA1c and a basic metabolic pannel.
Being/having particually underweight, have stringly hair, grossly overweight, have severe strech marks, suffer from severe and utterly debiliating fatigue, atypical psychotic episodes, pre-syncopy/syncopy on standing (othostatic hypotension), excessive thirst, dizziness, vertigo, heat intolerance, family history of similar symptoms, family history of endocrine of genetic disorders, myadrias, medically unexplained pain, use of endocrine drugs, moon face, insomnia/hypersomnia, abnormal sex drive, repeated infections, myalgia, athragia, cognitive dsyfunction noticable by others, hypercholestrolemia, abnormal appetite, salt cravings, low or high blood preasure, cold intolerance, abnomal bowl movements, hypermobility, easy brusing, body tanning...a good mix of these is an indicator of an endocrine or neurological disorder.
A full list of symptoms would be useful.