Cincou,
You are in a difficult predicament and I am so sorry that you have had to deal with these health issues with your husband and children.
I am not and probably will never be in your position, as my husband and I both suffer from some fertility issues. After reading the description of your continuous mucus cycles though, I can sympathize. I too have continuous mucus. And like some of the previous posters have suggested, I went to the Pope Paul VI institute (which uses the Creighton model of NFP).
They have trained fertility consultants all over the country, so there is probably one near you. Though my DH and I are using the method to achieve pregnancy, there is no reason why you couldn’t use the method to avoid it, even with continuous mucus. One of the reasons why the Creighton model is great for people like me is that the doctors who specialize in using it to treat people can identify the actual cause of the problem (what is causing continuous mucus). In my case, I went from having periods once every six months, or continuously for 3 months, to having a close-to-normal monthly cycle. There is no doubt in my mind as to when I am fertile and when I am not because the Creighton model really gives you the tools to know what is fertile and what just looks somewhat fertile.
If you are happy with your current method of NFP but still resistant to contraception, another thing you might consider doing is investing in a fertility monitor. The fertility monitor costs about $200 but can be purchased on ebay for much less. It is easy to
use and makes identifying peak fertility quite simple.
I cannot take issue with you for considering contraception as none of us can really understand the gravity of your situation as we are not in your shoes. You know and understand the Church teachings as well as I do and I am confident that you and your husband will make the decision that God intends for you.