Before getting on to the cat issue, Purrlishious, could I suggest that you have a couple of blood tests done by your gp? You've probably had them done already. One would be to check your thyroid function, the other your complete blood count. Some of your symptoms suggest that you may have a blood-related condition, or maybe be hypothyroid.
The cats. Keep them. There have been studies which have shown that prisoners or hospital patients fare better if they have a companion animal. I know from my own experience, that they can save lives.
In around 2005, I had severe depression. Because I'm epileptic and most anti-depresant drugs are counter-indicated by epilepsy, I had to wait almost a year before being treated - my gp wanted me to see a neuro-psychiatrist - during this time the only reason I was able to fight the desire to committ suicide was the love of my cats: Polly and Samantha (now over the bridge). Each time I had a paper scalpel in my hand one of them would come to me, purr in my face, and I just couldn't leave them to an uncertain future. Time again they would stop me on the brink. Luckily, the depression responded to the SSRI anti-depresant I was eventually prescribed.
Whatever minor physical complaints, such as the occasional flea-nip in summer, that we get from our cats, the spiritual benefit far outweighs. Humans have lived closely with cats for many centuries and they haven't wiped us out yet.
If you got rid of your cats, you'd most likely still be ill. Plus you'd be emotionally low, which would weaken your immune system. Cats are good for health, cuddle them and be glad.