Firstly wait till you are better before doing anything.
I think secondly you need to sit down with DH and discuss how you need to parent the same and agree not to scream/shout. Read various books on parenting after looking at amazon reviews. Discuss and use the best bits of the books together and agree a way forward.
Thirdly hold a family meeting. Agree new rotas, new rules - not screaming/hitting and out line what will happen in terms of discipline if rules are broken. How many warnings etc. do the kids have any ideas on how to make things run smoothly? Rewards etc. you may be giving too many chances/warnings and generally children will push things to the limit.
Lastly implement and be consistent. For things to get better, you need to make a lot if effort initially. There maybe s backlash initially but hold firm and ride it out.
A few things you can do. Stop the car if there is bickering or tantrums. Stop the car and quietly tell them you are waiting till they are quiet so you can concentrate on driving. And stop as often or as long as you need to. Wait it out. Be patient and late for school/clubs if necessary. Your ability to drive safely is most important. Get out if the car if you feel wobbly.
Tell your children they can have 15mins telly/a edible treat/treat in lunch box each day they are ready on time and with no squabbles. Sit them all down and run through a new timetable. Tell them you are not going to nag them but will tell them once only. Have a simple morning timetable on the bathroom door. Get a cooking timer. Set it at various times to go off. Ie) 7.15 to get up, breakfast 8, out 8.30.
The other approach is just not to get stressed about getting ready in the morning and not care if you are late for school.
They can earn screen time. So 10 mins each for various chores. Keep a running tally on your fridge. My children of a similar age sort the washing into piles/tidy their toys and bedrooms/empty the dishwasher/hoover/sweep. Once shown what to do they are fine. I always tell them we are s team and need to help each other.
Squabbles while eating - calmly remove the child causing issues and put them outside with coat/shoes. It's your house, you have cooked a nice meal and you expect children to behave nicely whilst eating it. Remain calm yourself and concentrate on catching up with how the kids days went.
Try and do some nice activities together and 1 to 1 - drawing /games/reading. Set an example if how to interact. It will make a huge difference. You won't look back in 20 years and wish you had cleaned more - thats for sure. You may wish you had given more attention to your kids though.