You are right. Putting your wife in a care home likely will be the hardest thing you will ever do. Unfortunately, there is no easy, right or wrong, answer. My
short answer is to follow your gut and allow your self to be with your wife as a husband, not as a caregiver. Unfortunately, this may be at the cost of not living in the same house.
Feeling guilty about your feelings and decisions is natural. You are very committed to your wife and this will not change. She needs you
and you need her. However, at some point alternative living situations may be necessary and it sounds like you may have reached that point. Sometimes we have to "give up" the caregiver role in order to be with each other as husband and wife. Your love for her will not change regardless of your decision.
A movie was made in the last few years called "Away from Her" with Julie Christie. The couple in the story faces a similar situation and the intensity of the difficult decision and transition is very clear. Given the closeness of your experience, I don't really recommend watching the movie. You may find it very difficult and personal. My point is that simply making a movie about this demonstrates that thousands of people face these difficult decisions and emotions (e.g., loss and grief) everyday. You are not alone and your love for her will never change.