OP I have been and regularly am where you are. I barely recognise myself with how terrified I am alone with the children and the weight of responsibility.
Some things which help me:
Distraction - housework, cleaning, doing dishes, cooking (something like a stew) very physical stuff that you can throw your weight behind
Watching comedy on TV
Doing one of the YouTube yoga sequences for anxiety
I find that when the sun comes up I feel better. So if you go on the weather app and see what time the sun rises where you are and then with each hour that passes you are closer to the sun rising and other people being awake and where you can get help if you need to.
I find that looking at other people’s Instagram stories is very helpful because it’s people all over the world in different time zones and they are awake and going about their day. And it’s distracting too.
Chatting with friends on wats app or on mumsnet.
If sick kid is really ill, having a physical barrier between you like a door helps so that you can mentally prepare yourself before going in to deal with them. Also having a plan of action is important so that you don’t panic and fall into inaction. Eg administering medicine, tissues, towels, drinks, calling 111.
If I think of any more I will add more