To answer your question first. Alzheimer's actually starts in the hypothalamus which is the part of the brain that control body temperature so it is possible it is the dementia causing the hot flashes. Niacin (vitamin B) can also cause such a flush. Actually there are a number of medications which have flushes as side effects. Low Estrogen can also cause flushes. Another cause can be low blood sugar. Perhaps you might want to check with her doctor to determine the cause since there are so many varied reasons for flushing.
Please do not feel guilty because you made a promise that can not be kept. There is not a person here, nor one I know, that wants their parent to be in a care facility. We want them health and at home where they belong. Or we want to be a super hero and be able to do everything for them. We make promises to ourselves or to them when they are healthy that we can not keep when they have multiple health issues. Then we beat up on ourselves because we can't be super human and do it all. With all of Mom's heath problems, complicated with your own health concerns, you did what needed to be done so Mom would get the best care possible. There is nothing wrong with doing that. No, it is not what we wish for, but it is what is best. Just be grateful that Mom is in agreement with the decision