How can I tell my family that I can no longer take care of my elderly mother, without them feeling like I just want to put her in a home?

after reading your posts and answer I would like to add a comment and share my experience. My mom was living alone in senior retirement. Occasionally my mom would tell me she had fallen in kitchen, bedroom, bathroom. Many times I was there she would say she fell, although she did not. The day she did fall, not broken thank god, and off to ems she went. She had been out of the hospital for 14 months. Even though I knew Nh was a safer place to be, I thanked god each day that mom was well when I visited her, I had a hard time knowing that I would decide to place her in NH. I would not unless I really really had to. Well the time came. It was the hardest choice. As I look back, I feel that the time came and it was a good choice. There was a long adjustment period,. I often realize that that fall was enough reason., before another fall would occur and mom may have been hurt, broken. My guilt for her safety needs would have been unforgivable. I did go and look at nursing homes, and chose the one who I felt that the staff comforted my emotions. I felt alone at times, but I know now that the lord was at my side. Sounds like you and your sisters are close and have been communicating. it's possible that the each had the same feeling you have but have not expressed it yet. preparing the paper work for poa etc is very important.
While mom is in NH, I find it now when I go to visit, it is a visit, I still keep in touch with her meds, and her progress, and I'm there for dr appointments. I bring home laundry and bring her snacks. I have two other siblings that relinquished moms care to me when she lived in Florida. I know it's a tough decision. Your thoughts were my thoughts before mom had her fall that headed her in the NH. My heart goes out to you. First and foremost, give each other a hug and I hope you will come to a conclusion and a safe haven for your mom. Take care and god bless.
Equinox