How do I handle every time I go to the nursing home my mom wants me to get her a new place to live?

Tell her you’re looking for a good place for her. Then drop the subject. She is on a loop, which is common for dementia patients. She won’t remember what you said.

If you put her in a new facility every week it would be the same thing.

Do you have any issues with the care she's getting in the NH she's in?
If not then leave her there.

Her doctor can prescribe medication to help with anxiety. Other than that there's not a lot that can be done.

Is she a religious woman? If she is then maybe you can have her clergy come by and give her a small cross she can wear or some other symbol of faith that can be a comfort to her.

It sounds to me like she's in a dementia loop. Does she tell other people in the NH about this man trying to kill her or it is only you?
If it's only you, then she may still possess enough cognitive ability to put on a "performance" for you. If such is the case, tell her thats not true and refuse to discuss it further.
If she tells staff members, then she really needs some medication to help her be calm.

Who knows what's going though her fractured mind? She may be remembering a scary person from her childhood or a character in a movie.
Ask for some anti-anxiety medication for her.

I think I would go that the police had been called that they found the man and he is now in prison in another state. You have to play into heir paranoia.

You can say..
You are safe here.
I told them at the desk not to let "the man" in.

I am working on finding you another place but I have not found any as safe as this one.

Keep telling her that she is safe where she is.

We’ve run into that too. While in her own home, she was certain people were after her. She is in her 2nd care home and was very content, but now believes I am trying to set her up to be killed or am trying to kill her myself. We have had to resort to a psych drug. The fear is not location based - it is in her head and will go with her. I’d rather my mother was medically calmed than scared.

You are certainly not alone. My mom has horrible paranoia and thinks the nurses and patients at the nursing home are stealing from her. I too am at wits end with this. I don't have a solution just yet but it's good to know that it's not just happening to me :}

When she brings it up, tell her you've notified everyone to protect her (to help calm her) and then change the topic. Keep changing the topic to something neutral, show her pictures on your phone, anything to distract her. She will keep bringing it up so keep redirecting the conversation.

I agree with others to talk to doc about anxiety meds.

One simple repetative answer may help. "we are working on it and it takes time."

Don't dismiss it entirely. My sister in law put her mother in the cheapest worst neighborhood imaginable. SIL didn't want to dilute her inheritance. The place had VERY seriously disabled "residents."

The sister in law was the designated caregiver, trustee and had all control.

Delusions and psychosis often happen for people with dementia. My husband had similar issues until his neurologist put him on Seroquel (generic: quetiapine) and the problems disappeared. There are many drugs out there with different doses. It will take some time to find the right med and the right strength. Once found, your life will be less hectic.

RE the NH situation, check out if she is right. Some places are nice when the family is visiting, but horrible behind the scenes. I don’t like facilities, so I care for my husband at home.