I'm a teacher, OP, and I was like you for many years. I ended up sleeping only 2 hours per night during the week in an attempt to keep on top of things, and inevitably eventually broke down.
You know where this is headed. If you don't take care of yourself now you will end up with a huge amount of time off for work-related stress at best, and give yourself health issues at worst.
You need to set boundaries for yourself.
To help switch off in the evenings, set a time after which you will NOT work, but use the last 10mins of your work time to make a to-do list.
Never have your work email on your phone.
Don't answer emails requesting non-urgent information at all. Wait until they ask a second time. 50% of the time, you will not be asked again.
Plan your lessons well. Make sure your paperwork regarding the children is up to date. Everything else is secondary.
Log the time you spend on a task during your PPA and after school. When you get asked why xyz isn't done, present your work hours and ask where it would have fit in - I do this all the time now and 50% of the time I don't get asked anymore, while the other 50%, some time magically appears. Say no to any extras - those are for bright-eyed, bushy-tailed young ones without families or those who are relentlessly trying to climb the greasy pole.
Exercise within the last hour in the evening or the first hour in the morning. It relieves a lot of worry and tension.
Your family need you and if you don't look after them and yourself, there will be issues in the future. They come first. There is one day a fortnight I have family commitments, which come above everything else. I have made is explicitly, and repeatedly, clear that I will not compromise on that. I refuse to do after school shite that day com rain or shine, and I refuse to partake in meetings - email me the minutes instead. But I was very loud about it until my school accepted that no really means no.
Working all hours and thinking about work all the time is not sustainable. I've been there, believe me, and it was a long road back to feeling like a normal human being.