You are not a bad person. At all. Even when you love your parents, when the mind starts to go, it’s a tough gig. My father had dementia, his four children and home care to care for his so he could stay in his house. I lived there and it terrified me to have to go home to him after working all day because I never knew what he was going to do. But I did have back-up.
God bless you and your husband. I don’t know how he does it by himself, all day. And I don’t blame you for the fact you resent them for taking him away from you. I assume you already know that caretakers sometimes go before the person with the disease, so don’t feel guilty about saying when.
As you said, life is not fair, so if they find it “unfair” when you finally put them in a home most likely paid for by the state, remember that. Life is unfair for everyone, even those with Alzheimer’s.
Lastly, you can’t help how you feel, only your actions. And your actions are commendable.