This might not help, but anyway... I was 21 when my mum died, after 9 months of knowing the end was coming (cancer too) and 22 when my father died, a stroke, so no preparation. I totally identify with your comment about feeling that your dad died already, because you grieve so much during this waiting period. Having gone through it both ways, in such a short time, my feeling is that you do the same amount of grieving over a different amount of time - does that make sense? I mean in that, with my mum, it was protracted, and we watched her go, but got much of the grieving done, without realising it, over that time, and adjusted more quickly afterwards. With dad, it was a huge shock, and the grief was much more intense, deeply painful, afterwards.
I was scared of death. I've tried hard since then to teach myself that it's part of life. We fought it, denied it, with my mum. I wish we could have found the peace to accept what was happening and enjoy the time that was left, rather than railing against it and denying it. If there was one thing I could wish for you it would be that; to understand that, however painful it is, you need to accept that it's happening and somehow make the most of your dad's time with you.
As for what you tell your children, I can't speak from experience. When our cat died (no, it's no comparison, but the only experience of death they've had) we explained it simply and honestly - that she was sick, and couldn't get better, and that she didn't have long left. They were upset, but they coped remarkably well. Let them believe in Heaven if they believe, or not if they don't. Take your cues from them. My DD'd teacher recommended the book "Badgers Parting Gifts" and it might be worth looking for it.
All the best, my sympathy goes out to you, it's a tough time and I hope you find the support you need.