Hi Jo - your symptoms are all highly characteristic of hypothyroid, I am having a similar battle with my doctor because my thyroid results are 'normal' but i am still symptomatic - I got much better when I first started on thyroxine but my doctor wont increase my dose past 50mcg because of the normal test results despite my symptoms. Apparently the 'normal' levels are quite controversial and there are lots of different views about what is a normal level.
A lady on here called Maggie has just recommended a book to me by a consultant who helped her get her thyroid sorted - Understanding Thyroid Disorders by Dr A Toft. I shall be buying it to get the info I need to go back to my Doctor!
However ..... although your symptoms do sound very like hypothyroidism, they are also going to be symptoms of lots of other disorders and problems, thus confusing the picture for your GP!!!
Your hand/arm problems sound like Carpal Tunnel Syndrome which can occur with hypothyroid - but equally can occur spontaneously, or through other reasons like RSI, nerve damage, water retention or arthritis. (I sympathise - I get it too. I used to wake up in the night with an agonising burning numbness from my shoulder to myf fingertips. I would have no feeling or mobility in that arm - I would have to use my good arm to shove the dead arm off my body in order to be able to slide out of bed - it was like pushing someon elses dead, amputated arm off me, the most disgusting feeling!!!!) I find splinting at night is helpful when its really bad - you can get some professioal ones that have a metal bar on the inside of the wrist to keep your hand in the right position.)
Weight gain can also cause CTS, muscle/joint pain, breathlessness etc. Exhaustion and weight gain can be like a vicious circle - too tired to move so you put on weight which makes you even more exhausted - and on and on. Im in the same problem myself which in my case is exacerbated by severe osteoarthritis and degenerative spine disease. Which means I am on lots of meds (along with the bipolar disorder I also have) which leads to.....
Medicines side effects - are you on any other meds at all? They can often cause side effects like weight gain, tiredness, nausea, muscle pain. Some can cause poor immune system - eg sore throat and glands, anti-depressants are especially known (certain types) to have a risk of this?
And finally - are you really sure you dont have early menopause? The night sweats, tiredness, nausea, tinnitus, cold extremities, tearfulness - they sound very like a hormone problem, and 39 would be early but not exceptionallly so - its very possible... Have you had regular gynae check-ups?
With hypothyroid, you tend to feel freezing cold all over, all the time - not just hands and feet, and not night sweats from what I have read/experienced - so that would fit better with early menopause, perhaps?
Or these could all be the physical symptoms many people get with moderate to severe depression: not imaginary or psychosomatic: depression the illness causes physical symptoms as well as psychological,emotional and dysfunctional.
Keep going back to your Doctor until you get it sorted. I learnt the hard way that Doctors arent psychic and dont know your still feeling poorly unless you tell them. And will assume a medicine is working unless you go back and tell them otherwise!
Ask for another set of thyroid tests - not just one, there are several tests including T3 T4 TSH and antibodies tests. Ask for a complete blood scan for infections, liver function, kidney, diabetes, vitamin/mineral deficiencies.
Ask for a hormone test to establish whether you have low levels or are having an early menopause.
Ask him about the possibility of trying a low-dose mild antidepressant while all these tests are being pursued. You cant go on feeling like this whilst tryng to cope with 2 small children - its going to wear you down. Sometimes antidepressants are prescribed for purely physical illnesses - for example unexplained, incurable skin irritation and itching. For some reason not fully understood, some physical symptoms are considerably eased by anti-depressants: at least have the conversation with the doctor and see what he or she says: you need something to help you feel better soon, until you know exactly what the cause may be...
In the meantime - do everything you can to make sure you get a good 8 hours sleep every night, decent nutrition, rest as well as sleep if possible - I would say while yr body is under this much pressure it is best to listen to your bodies messages about resting rather than exercising etc. You can start up again once you have got to the bottom of whats going on. Be kind to yourself in the meantime - early nights, peace, quiet and calm, eat well, get the support of those around you. I really hope you find out soon what the problems is and start to feel better.