I’m currently no contact with my mom who needs home help but living a rural area, there are not many caregivers close by. So the caseworker has only been able to get her 3 hours of help per week. I know the case worker visited her last week and I’m hoping she talked to my mom about assisted living. If my mom is willing, can the case worker find her a place to go that has an opening and will accept Medicaid? Or do I need to be doing the searching?
Your Mom could go into a faith-based facility if there's one in proximity to her. Most of those are more affordable than a "chain" facility or a for-profit one.
The other option is for her (or your sister) to privately hire an aid to come every day and check on her. She would need to cover the cost for this herself. Your sister may need to help her manage it. Or, she pays a trusted neighbor to check in on her daily and report back to your sister.
With folks who are "stubborn" often they will respond better to someone in authority telling them what they must do.