How do you walk through moving out of state from parents with dementia living in assisted living and sister who controls everything?

Tell parents and sister you are moving. Tell them you will continue to visit by phone and that you will continue to come to visit when well enough. There is nothing else that needs to be said. Your responses to your sister will depend upon her response to you. If she is supportive then you may get along. If she is not supportive you may not. If you don't then you may hear less about your parents other than what you hear directly. If you sister needs help she may need to hire a fiduciary to help manage your parents.
Your own needs need to be addressed. You have decided this move will work best for you. Make it. That doesn't mean people will not be sad/angry, whatever they choose to be. But you must get on with your life. Respond gently that you feel sorry that they are so upset, but that you must do what you think is best for you. If they choose to feel you are selfish that is up to them. You cannot control the feelings or actions of others. Just your own. I wish you such good luck.

If your parents are in a facility receiving good care, what is your sister upset about? What else have you been doing to contribute to their care that your sister thinks cannot be covered in another way? Can you tell us what she wants to happen besides you not leaving? I don't know if there's a magical solution where everyone goes away happy but IMO it's not reasonable for your sister to hold you hostage there with her immaturity. Move and let the chips fall where they may. She will hopefully come around when you leave and the sky doesn't fall.

Sister has DPOA for health and finances and wants less responsibility? Are you able to talk over your differences and perhaps come up with a way to compromise? Does sister understand your physical problems? Does she want to move to a warmer climate?