I think your doctor knows as you've said there is no pill that will give you the good quality sleep you need, they're just a chemical sledgehammer :/
The CBT sounds a fantastic idea, it's actually great that he is willing to ensure you rid yourself of this anxiety and not simply mask it. You may not relish the idea of going through this but the problem isn't going anywhere is it?
I would also see if there is anywhere in your area that does guided meditation, that is fantastic for deep relaxation. Self hypnosis is great too, I used that to stop smoking and drinking (not at the same time) concentrate on your heart rate slow and steady initially a few minutes a day I'll PM you a couple of techniques if you want.
Is the anxiety only associated with the night time? As said have a few cat naps it may retrain your brain to actually associate sleep with total relaxation. It makes sense your health issues now too Ruby as we heal when we sleep so if you rarely sleep you're at a distinct disadvantage.
Take your doc up on his kind offer you've nothing to lose, it might be playing the long game but you know being drug free and in charge of your own sleep pattern is the best option in the long term. x