Having a sick loved one, or getting this kind of bad news ... it is one of the most difficult times of life and relationships. Your reaction of panic, dread and the like is so expected and universal that it is truly the "natural" reaction. It was a shock, your mind fills with all the "worst case" scenarios, dread and sweating and depression ...
Our Zen Practice is not about never losing our balance (some folks think that Buddhism is about that, about never reacting in a "human" and "negative" way, but I do not think so). It is about being able to return to our balanced, center point more easily after life knocks us for a loop. It is about getting back on the bike after we fall off. Thinking that a loved one will have a serious illness will sure knock someone off the bike.
This is one of those times when I believe that Buddhist Practice is recommending something very different from the "either/or" way of looking at things that most of us think is necessary (e.g., either I am attached or I am not, either I am happy or I am sad, either I am calm or I am worried etc.). Instead, we can be both (or many things) at once.
So, for example, my attitude is that we should be completely in love with our loved ones, throw ourselves into it without reserve, savor each moment of time with them, do not resist that ... yet be willing to release them when the time comes. This is "being present in the moment" while yet "not being attached" or "clinging" to that moment when it ends. It is like looking at a beautiful scene ... a painting made of chalk on the ground, for example ... just appreciating it while it is there, but allowing it to wash away when the rains come. (Is that, perhaps, a kind of "unattached attachment?")
We can also be "happy" "sad" "grieving" and "peaceful" all at the same time. When faced with a dangerous situation, we can even learn to be "worried" and "calm", "accepting" and "not accepting" simultaneously (trust me, we can!). So, for example, my first teacher, Ikuo Azuma Roshi of Sojiji, lost his wife after I had known him a few years. For many weeks, he was not himself and was easily a bit teary eyed. I was SHOCKED because, of course, Zen Masters are supposed to have surpassed life and death and all such petty human emotions. So, as I had known him so long and we talked about anything and everything, I asked him about this, "If life and death are states of mind, why are you upset?" He said to me, "Life and Death are nothing; I am sad because wife die."
That shut me up. He looked at me like it was the most obvious thing!
When my own mother died a couple of years ago, I was sad, I missed her ... I was also amazingly happy. In fact, I threw a funeral for her that was more of a "birthday party", saying to my relatives that "I am just as happy about ending the show of life as I am about celebrating the start". They thought I was a bit crazy (maybe so ... they are used to me by now), but that is how I really felt.
And one more thing: When we get bad news it is natural to want to run away. Sometimes we should, like when the house is on fire. Sometimes we need to stand our ground. Finding your balanced center will help you make that choice. And being present with a relative during a long illness is (not only something almost none of us can avoid sometimes in life), but is a gift ... the true battlefield where our Buddhist Practice is put into action. It is a time for a great maturing in Buddhist Practice.
It is important to know that "being home" is not a matter of "staying" or "going". If you leave, that is where you are. If you stay, that is where you are. Sometimes it is best to go, as with that house on fire. But sometimes if you go, you may or may not find the peace that you would have found by staying. Only you can know which is the right choice in your life, and Buddhism does not fill in the details of the choices we must make (only your own heart can do that).
As to Buddha, Dogen etc. having something to say directly on this subject ... Buddhists of old just left their families at home and moved into monasteries with the boys!. :-) So, this is really something that they did not write about very much. As Buddhism has come out into the world, there are many new situations to which we must apply very old Wisdom and Compassion.
Did that help?