Hello esinedee, yes, many of us have been there. I'm sorry you are feeling terrible. I think that even when there is no alternative, that it's probably impossible to "feel good" about moving someone into a care home, for a variety of reasons.
My mother was a little less advanced than your mother, and with it enough (well, some of the time) to know that she was in a care home and not in her own home. She had been in hospital for about 12 days prior to the move to the care home, but had no memory of that, nor the crisis that prompted her being sectioned. (She has Alzheimer's and short term memory problems.)
I got all the blame and it was not pretty (the usual accusations of wanting to steal her money and so forth). Like your mother, my mother was "fine" living alone at home, there was nothing wrong with her, this was all my fault, et cetera. In reality, my mother was not fine, her flat was filthy, she wasn't eating, she wasn't taking her medications properly, she was malnourished and underweight, she was wandering and getting lost, she had given money to scammers, I'm sure you know the story.
What my husband and I did, after the day she moved in, and with the support of the care home staff, was that I didn't visit for about 6-8 weeks. She responded well to my husband and he would visit her. When I did begin visits, I never went alone, and never stayed long.
I think my mother settled fairly quickly, definitely within the first few months. We did get a lot of upset and confused phone calls at the beginning, but those have long since stopped. I would say to give it at least six weeks before you begin to worry, and I'd also be sure to talk to the staff (privately in their office or on the phone) often to get a sense from them, about how they feel your mum is settling. Even though I expected it with the dementia, my mother's version of how she was (I sit alone in my room all day and I don't have any medicine and haven't eaten and there is nobody here) was very different from the reality (your mother is happily playing bingo/chatting in the lounge/has just eaten ice cream).
Try to be kind to yourself, and maybe you will sleep better, knowing your mother is safe and looked-after. I found it took me a few weeks to get used to not having to worry about her every moment of every day. It's a big adjustment for everybody.
Best wishes to you, and if you get a chance and are so inclined, please feel free to update us.