Some of it does sound a bit like a lingering virus kind of thing - the aches and tiredness and such, especially if you had an illness like a virus, bad cold, flu etc... before this came on. I know it's taken me a while to recover from somethings like viral pnumonia and a virus which made my brain swell up. It can last a while. I remember Em10 I think having a similar sounding problem a while back, though some different symptoms.
Things like fibromyalgia and chronic fatigue syndrome type illnesses can also be kicked off by some illnesses.
Could the dry itchy cough be being triggered by something like some kind of allergy or irritation? Similarly the headache - I have had constant headaches caused by things like switching to artificial sweetners and a Dianette contraceptive pill.
I agree that going in demanding to be tested for those kind of illnesses may not go down well! GPs will be looking for general indicators of things which could lead to suggesting, with your symptoms, ideas as to what kind of thing might be wrong and then any referral as appropriate (like the ENT one). I don't think GPs would order a headscan or that kind of thing??
Thinks like Addisons and SLE would, I'd have thought (I know people with both) be initially investigated, diagosed and dealt with by specialists. Thyroid is something you could ask about though if not already done.
Sounds like something a specialist at the hospital would do if they feel it would be useful. ENT referred me for a head MRI once I was referred to them.
Are you on vitamin D therapy and did they say what your level was? Whether it is a proper deficiency or whether your levels are a bit lower than normal? My rhumatologist says deficiency can make you feel quite run down when I've had very low levels.
Do you know what blood tests you have had done and anything above the usual (full blood count etc...) like inflammatory markers or an ANA or thyroid?
Try breaking things down and try and work out what is affecting you the worst and try to tackle things individually as things as a whole can become quite overwhelming.
Unfortunately some things just (referrals, getting better etc...) take time, I hope you feel better soon though.