I'm so sorry you are going through this. I have absolutely been there when my Nan was dying (we were incredibly close and I was doing a lot of the caring/dealing with social care/running around etc. ).
It's absolutely brutal on its own and our job doesn't give any breathing space!!
You have had some good advice already, but some things that helped me/I wish I had done;
Please talk to your head or another member of SLT and let them know what you are going through. I literally turned up at the head's door in pieces one morning - although she couldn't fix it, she was kind, understanding and supportive which meant the world to me.
Can you tell colleagues in your department? Again, I told a couple of close colleagues and their support kept me going.
Accept all and any offers of help.
Please cut yourself some slack too! Plan a couple of lessons a week that are easy to deliver - the kids will be fine. Keep marking to an absolute minimum and do as much as possible in lesson. I prioritised marking for Y11 and 13, all other classes, I did feedback in lessons only.
Same outside of school (not sure if you have kids etc). Easy meals that require minimum prep and effort - a ready meal or something on toast won't kill anyone. Housework - do what absolutely needs doing and nothing more.
It's so, so tough going through it so make sure you take care of you ❤