Why is Grandma is saying horrible things about me?

Almost 4 months ago, I moved my Grandma in with my family and I. She wasn't able to care for herself anymore and she was very lonely. My husband and I pay for her living expenses, my uncle and my mom pay for some caregiver help. She is pampered and VERY well cared for. We take her to get her hair and nails done, she is taken out to lunch almost every day, is always clean and well fed, never left alone (which is incredibly hard on me, I am a busy working mom of 2 very active teens and wife to a firefighter that is gone for days at a time), yet she bad mouths me every chance she gets. Especially to her caregivers. Usually she tells them that my house is actually her son's and we are freeloaders (not true, I own my home), that we starve her and her cat (they are both obese, and they both eat nonstop, not true), that I beat her (never), that we lock her in her room (never, but sometimes I want to). Yesterday her caregiver told me that she has been telling her that I am cheating on my husband (NEVER!). She is very confused at times and has never been diagnosed with dementia but she definitely has it. She goes in and out of lucidity. When I talked to her about it she denied it (I have heard her say these things myself, I know she says them). She cried and yelled and went on and on about how people are making up stories about her. It was awful. I was very understanding and kind when I confronted her, I just said "Grandma, I know you get confused and mixed up sometimes, but it hurts my feelings when you tell people untrue things about me. I know you don't remember saying it but you did. I am hoping that by us talking about it, maybe next time you will remember to only say kind things." It as very thought out and practiced, I didn't want to attack her but I felt it needed to be addressed. I am at my wits end. She has become a mean, miserable person and I feel like I have trapped my family. She is rude to my daughter and complains about EVERYTHING, constantly. I was trying to be kind and loving to her, but i don't know how much longer I can take this. My home is no longer a place I enjoy. Is this behavior a typical progression of dementia? Is there anything I can do? We have already tried Xanax (for her, although I could probably use some too), it helped a little but not enough. Please, any advice would be appreciated. I am trying to keep her here as long as I can but if this behavior continues I will move her to a home. I refuse to put my kids through this any longer.