Nothing to be ashamed of, Abe, any more than if you had a broken ankle or, like me, rheumatoid arthritis or some other horrible disease. And I wouldn't be so sure that the people you see are strong and positive. Up 'til fairly recently, people would comment on how well I was doing, having gone from being pretty active to having very limited mobility, and I would laugh and joke with them. Then come home and cry and cry because I wasn't coping at all. My OH was worried sick about me and didn't know what to do. It was only when I admitted to one or two people and on TAS that I wasn't coping very well, that things started to improve - not physically unfortunately, but mentally.
The thing is that I was doing what so many people do, and putting on a face for other people and I think it made matters worse because I had the strain of acting a part as well as having to come to terms with having a serious illness.
You've made a good and important move by admitting to us on here how you are feeling. Do go along to the CBT sessions. I'm sure that you won't have to say any more than you are comfortable with and will be able to go at your own pace.
You don't mention a partner, so are you bringing up the children on your own. That's quite an undertaking. You are clearly a loving dad and they will pick that up. They are lucky to have someone who, although ill himself, is more concerned about the effects of the illness on the children than anything. That is something to be proud of.
If you want to PM me any time, please do. I'm no expert but I can be a virtual listener.