Beating a boxing bag to pulp or lifting heavy weights helps me specifically, but all exercise helps because of the endorphins. I don't know if CBD supplements are legal where you live but they also help keeping calm. IMHO that's just dealing with the symptoms though, which to me sounds like you're stretched too thin while people keep demanding more and more and MORE and your have nobody to help you with your stuff.
Can you say no to some obligations / things? How involved are you in this family war? Do you even want to be involved? How does that articulate itself? Do family members vent at you about others or try to get you to take sides? If so, I'd honestly block them for a while (not forever unless maybe you realise that Aunt Meredith is a right old
witch and you don't need that energy in your life), or put their messages on mute and archive the chats.
I'm guessing there is nothing you can do about bringing peace to this war. If that's something you're feeling like you should be doing, I suggest you stop trying. What's the worst case scenario if this war escalates? Are there any consequences for you specifically if things come to a head? If not, wash your hands off them and let them duke it out. It's none of your business what other, grown adults are warring about.
And your mum - do you need to run when she asks or is it some sort of guilt button that's been installed in you? If she's in hospital, her needs are taken care of. She won't starve, she won't get hurt. How much time do you want to dedicate to her? It's ok if you say "-5398439 hours a week, please give me back my life" btw
Realistically, she's a grown
bleeping adult and shouldn't need you to run her life for her. Yes, being ill sucks, but so does being run off your feet until you explode. If you feel super guilty for "leaving her", make it a managed thing. Dedicate 15 minutes to her demands in the morning, write a list of what she wants and manage her expectations to the best of your ability. You will come by at X time on Y date and you have Z hours / minutes to stay by her side. You will run X errand at that time and not earlier. Be very very specific and clear and if she needles you with "but why can't you do it earlier etc bla bla" ignore it. Repeat your statement and become immovable. This is your life after all.