My mother is moving from an assisted living apartment to a SNF/nursing home soon. She's in rehab now in a room with 2 others. She thinks the rehab set-up is temporary, and that she'll be moving to another room with only 1 roommate once she finishes rehab. I sure hope so for her sake.
I can't even deal with the fact that elderly people have to move into a small room with a complete stranger and share a bathroom. A nursing home near us made major modifications to have one person per room with a private bath. Now you're talking!
Keep looking. (I sure hope you haven't paid any deposits?) And be cautious of verbal commitments; get any changes or issues on numbers of people in the rooms in writing.