How do I move on now that I've lost my father too and ties with my sister are strained?

Annie2019–
Very sorry for your loss, especially back to back. I will be losing my “caregiver” role soon as well and am pondering many of the things you are. Difficult relationships with family, resentments and moving on are all reasons not to get out of bed, yet we must find will to move past all the things we’ve been through so we can build onto our new life. People suggesting allowance for grieving are right, but how do we know how much we will stay there? Seeking to care for others by volunteering is also right; I long to serve others in a easier way than this 24/7 hell loop of caring for one. Therapy? Most definitely, and why not when there’s conflict with family? Every one of us must determine for ourselves the best course of action but sometimes it’s the lack of action we need most. To remember what it’s like to be still and as untroubled as the deepest pond in the forest. To realize what it means to have autonomy of ourselves once again, to just be our own caregiver. I’m struggling today and will tomorrow no doubt, but I had to let you know, I see you.
Big hugs from afar, sending peace to your heart.

https://www.agingcare.com/questions/my-sister-and-i-are-caregivers-for-our-parents-the-responsibilities-are-not-evenly-divided-between-u-445911.htm

The above post was made in 2019. You say in the post that you work full-time, sister part-time. That you expressed to your parents you could do more but "They" kept relying on sister for things like appointments. This is not you fault. Seems to me, from what I read, you really tried to be there and do what you could. Its not your fault that your parents relied on her more. Really not sure what more you could have done.

First, you need to let yourself grieve. Let your sister grieve. And I agree with Alva that you may want to talk to a therapist. And I say this because of this phrase.

"Now I have lost my caregiver role and the focus and center of my life." 

You need to find "tools" to get you out of this mindset. Your caregiving days are over. You need to find something to fill your days. You have work. You now have time to make friends. Do things you couldn't do before. Find a new you.

Firstly, I am so sorry for your loss. It is the progression. We lose our parents and have usually a good deal of life to live beyond that. You have a whole lot of insight into the fact that your primary identity was as the caregiver. I was a nurse, and I know the feeling. There is a strong sense of being of worth and of value when we are caregivers.
You may, in all honesty need therapy to help you to move on, explore options. You will be out of your comfort zone in reaching out the the world and making your way in it when you have in some ways been "cloistered" in the caregiving roles. Some Licensed Social Workers work on their own doing this therapy; they are LCSW and clinically certified in family counseling, in grief counseling, in life transitions counseling. They are often as good as, or better than more expensive psychologists. Give that some thought.
Know that there is no hurry. Right now you are grieving.
If your Sister and your role was that YOU make amends normally, and she "forgives", that may be a relationship that needs to change, or to be pulled away from and replaced with friends with whom you can be yourself, be comfortable in being yourself, be honest, and not feel you are the "one who seeks forgiveness". There came a time in my life when I had to sit with the knowledge regarding a very close blood relationship that, in all honesty, she may not be a very nice person. I can't know the details there, but that would be another thing to work on and comb out.
Those are just a few suggestions. There's nothing wrong with being a caregiver, and there are many roles out there in which you can volunteer, from fostering puppies to taking therapy animals to shut ins, to visiting in Nursing Homes. You will meet others as you prepare to do this valuable service.
Consider a younger you. What were the hopes, the dreams. Our society is opening up. We have all been shut down and we are ALL a little uncomfortable. It's almost a "ggod time" to be on an exploration of self. I wish you such much luck. It is scary to step out of the comfort zone, but it is also exciting.

You can't change your sister's attitude until and unless she comes to terms with herself and moves on, so set that issue aside and focus on yourself.   Sometimes the more you try to address a hostile, the more the other person is aggravated.  She may have some issues of her own that she needs to work out, by herself.

As to your the loss of your father, I'm sorry to learn that someone so close to you has passed on.    But think of the "caregiver role..., focus,... and center of your life" in terms of other people and areas of need.   Leverage what you've learned by continuing to provide care and love to others.

Have you done any volunteer work with scouts, with charities, animals, or other people or agencies?    What special interests did you have before caring for your father?    Would you be interested in volunteer work with other older people, such as delivering meals on wheels to homebound folks?   What direction would your life have taken if your father hadn't needed care?

And where do you want to be in a few, then 5 or 10 years from now?   What was your profession before you became your father's caregiver?  

This kind of extension of caring by reaching out to other areas in the community is what helped me through some dark times; I haven't acted on the plans, but the plans themselves are excellent motivators.

But recognize as well that you've experienced a traumatic event and need down time to heal as you move forward.   Give yourself time to grieve, much longer than 4 months if you need it.   BTW, why is your sister so angry?   What changed 3 years ago?