how in the (heck) does one carry on?

As best one can. There really is no alternative, in the end. Everyone will sooner or later go through the same situation, and the lucky ones get to say goodbye first. It tears you up inside for months, and it will stay with you for the rest of your life in one way or another, but death is a non-optional part of life and part of growing up.

I went through a period of a few years in my teens where I was attending a lot of funerals, losing about 70% of the relatives on my mother's side of the family in a very short space of time. When I was 29 my mother herself died of cancer at the ago of 50, after a fairly short period during which we knew it was coming. Even so, the end came a lot faster than anyone expected, and it was only good fortune I was able to get there in time to spend the last few hours with her.

Even now, 10 years later, I have the occasional emotional flashback to the grief of that time, triggered by the strangest things that bring her particularly strongly to mind, although she is always in my memory. I miss her.

The point is, you can't let the inevitability of death ruin the enjoyment of life. It happens to everyone, and the only defense is to make the best of the time you do have with people you love. A close friend of mine went through something similar to you with his mother a few years ago after a cancer scare, and I gave him the same advice I'll give you, for what it's worth: Make the most of what time you have left, don't dwell on future events to the exclusion of present ones. His mother is still going, as is your father, and the best thing you can do is to make the most of what you have now. Build up memories you will treasure later, because believe me your loved ones live on that way.

pca