Of course it is not just about the DCs @Carrie , and we are all here to support you
So my Dad died ten years ago but you are right, my parents divorced when I was at university. I kind of mourned our relationship at that point too. For example, at my 21st (2 months after my Dad walked out)I remember thinking we shouldn't start eating because not everyone was there - but then realised that all the guests were present, it was my Dad that was missing.
I am not surprised that you are struggling to adjust. I think that your Dad's health deteriorated very fast. Whereas for me my Dad was diagnosed with incurable prostate cancer 2.5 years before he died and then a few months before he died we learnt that it had spread to the brain. So I had a lot longer to come to terms with what was happening and it was a more gradual letting go. The last couple of weeks of his life were painful to witness.
Easier said than done but just try to take each day as it comes and expect a bit of an emotional rollercoaster. I also wouldn't worry about the DC seeing you cry - I think it will help you all. My DS was only 22 months when my Dad died (and I had only been his Mum for 6 months). Looking back I probably overdid it trying to keep everything normal for DS. But when my Gran died aged 100 a couple of years ago I did more openly grieve in front of the children.