How to cope with anger and depression?

It's hard to do 24/7. There's no reason that you're delivered this task and be further burdened by *ssholes & *sshole words......
Take charge of your moms care and estate. Take charge and tell your siblings what to do. Take charge. Don't ask, tell.

Your mother will "consider" Assisted Living once you tell her you quit your inhumane, 24/7 365 day per year job caring for her! Think of it like this: what would she do if you died on the job from stress? And dont think thats a joke either. My friends mother was causing her SUCH stress she was getting high blood pressure and chest pains as a result and had to be put on meds. Her mother died unexpectedly last week and she was off of her meds within 4 days. True story.

Your mother would do either of 2 things if you weren't around (God forbid): she would pay for in home care givers or she would move into Assisted Living immediately. Instead, she's treating you like a slave and your sister like a queen, demanding you apologize to her for expressing frustration over being justifiably burned out. Stop allowing your mother to take such terrible advantage of you and put your foot down right away before you become a statistic! And tell your sister you'll apologize when SHE'S walked in YOUR shoes for a month. Until then, she should be grateful you even speak to her at all.

Wishing you the best of luck moving forward with your own life now, you deserve to.

Why can't you leave and move on? Is it for financial reasons because you can't afford to go? Give us more back round about your situation.
Now, in most caregiving situations involving elderly parents and adult children, all of the responsibility and the day to day, hands-on 'dirty work' gets dumped on just one. It's unusual for all the siblings to help out.
You're not alone in that.
Next, you are an adult. No one can make you apologize and you should have never allowed your mother to reduce you like that. It doesn't matter if you live in her house. It doesn't matter if she pays you to be her caregiver.
No one (even kids) should have to apologize if they aren't sorry and if they didn't do anything to be sorry for.
It's sounds to me like your sister who does nothing to help with the care needs had a telling off coming. Don't apologize again.

You have to realize that you, one person can not be a 24/7/365/ caregiver. You need to have caregivers that will come in and help out. These caregivers are to be paid for by mom not you. You get the caregivers for how many hours you think you need help and for the hours that are the most difficult for you. If you have been expecting your sister to help she has pretty much show that she is either not interested OR she has set boundaries and she idls sticking to them.
Your other option would be appropriate placement for mom. IL, AL, or MC.
You do not say what type of care mom needs or her medical problems so it is difficult to say what placement would be safest