How to do I deal with an elderly mother's manipulation?

Just what I was thinking, don't tell her of your plans. Just tell the other sibling when u will be going and coming. They can call Mom once you are gone that Mom is to call them if she needs something. Block Moms calls while away. Tell sibling to please not call you unless an emergency. Everyone needs to enjoy their time away.

Don’t see her every day during normal days. Then, do NOT tell her u r going on a trip. See her just before you leave and then immediately whenever you return. If she calls while u r away, tell her u have been busy or have a cold and don’t want her to catch it.

Clearly she is frightened to be left alone. Do supply her with the 911 number and tell her to call when she is fearful or worried. She will likely do so, and will end in hospital; just be certain you don't return home unless you know something really is wrong.
If she has the other sibling you will just have to stick to your guns. Tell her you are leaving. Give her a calendar to mark off the days you will be gone. And go. You won't change her reaction. You have just to override your OWN reaction to her reaction, and take the needed vacation. Hope you have fun.

Aubonpain
I, of course, don’t know your mom but I can tell you that many elderly parents become attached to the one adult child and do suffer extreme anxiety when they know that child is out of town. I had a friend whose mom would actually check into the hospital when friend would visit her daughters family a couple of hours away. This woman was competent and very level headed on a day to day basis but just let her get wind of an impending trip for my friend and the worry would ensue.
Friend did not change her plans. Her mother didn’t ask her to but they both knew it would be very uncomfortable for the elder while friend was away. She had a great relationship with her doctor. He would give her medication to help calm her nerves.
My advice would be to tell your mom that you know she’s uncomfortable. That maybe you need to be away more often so that she would be more accustomed to your being gone. That you or your sibling will take her to the doctor before you leave so she can check everything out but you will not be changing your plans. If she thinks there is a chance she can convince you to stay, she will work on that chance. So be clear. No need for you to be upset. You can understand it’s real anxiety for her but you will be gone. If she needs to go into the hospital, that’s okay. Sibling will take good care of her. You will see her when you get back. Mean it when you say it so she can focus on calming activities instead of trying to make you change your mind. Good luck. Enjoy your trip.