How do I handle it when my mom says she wants to go home?

Get her seen by a geriatric psychiatrist. Meds can help with agitation and sundowning.

Can your mother arrange her move on her own?

If you said "that's fine mom, let me know the date you're moving out", how would that go over?

Can she make those moving and care arrangements herself?

Oops, ran out of room............
You don't have to spend tons of $$$ on a doll for mom either. Just wanted to show you the various dolls available and the articles on the subject with that link I provided.

Wishing you all the best!

Here's some tips for dealing with a dementia sufferer who keeps talking about 'wanting to go home':

If you are the family caregiver for a senior with Alzheimer’s disease, a phrase you’ve likely heard them say is “I want to go home.” In most cases the older adult is already home, but home is an environment that no longer looks familiar to them. While it might be that memory loss is causing them not to remember their surroundings, the phrase might also mean something else.

Communicating with a Loved One Who Has Dementia

Adults with Alzheimer’s disease and other forms of dementia often lose some or all of their verbal communications skills fairly early in the disease process. This makes it hard for their loved ones to understand what is wrong or what the senior is trying to say.  

Sometimes a person with dementia is searching for home because of unmet needs or because they are feeling isolated and alone. Here’s what they might really mean:
Undiagnosed pain: Is the senior in pain? Sometimes you can tell by the look on their face that something isn’t right. If they aren’t able to tell you what is wrong, try pointing to different parts of the body and asking them to answer yes or no if it hurts. 
Sad or lonely: The very nature of Alzheimer’s disease is isolating. As memory and abilities slowly decline, the older adult’s world becomes smaller. They might remember faces of loved ones but be unable to recall names. Eventually, they live in a world where no one and nothing looks familiar.  
Confusion: As short-term memories fade, an adult with memory loss might be confused about where home really is. They may try to find their way back to a house that was home many years ago.
Unmet needs: Sometimes the senior wants to “go home” because they are hungry, thirsty, tired, or need to use the bathroom. When the environment they are in doesn’t look familiar, they feel an urgent need to search for home. It can help to offer them something to eat or drink or show them to the bathroom. 
Boredom: Creating meaningful days can be difficult when a loved one you are a caregiver for has memory loss. Their need to find home might be the result of boredom and not enough activity.
Redirecting a Senior with Alzheimer’s Attention
When a senior with Alzheimer’s is frustrated and agitated at not being allowed to search for home, redirecting their attention is often the best solution:
Ask for their assistance. Bring a basket of towels in to the living room and ask for the senior’s help folding them. Or try offering them a dust cloth and asking the senior to clean the dining room or help make dinner. Giving the senior a job may help redirect their thoughts.
Turn on favorite music. Research shows that music has the power to change mood. Uplifting songs can boost mood, while the sounds of nature can soothe agitation. Music an older adult associates with the happiest times in their life might also help calm the senior and make them forgot about searching for home.
Agree to go later: If you aren’t able to redirect your senior loved one’s attention, agree with them that you will go to their house later. Tell them you have to make a cake and clean up the kitchen first. Agreeing with them might help soothe the agitation they feel, while also buying you time. It’s entirely likely a senior with memory loss will forget they were searching for home before long.

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If she's getting increasingly agitated, I recommend getting her a baby doll to hold, rock & swaddle. Many, many women (and even a few men) with dementia LOVE their baby dolls and find great comfort in taking care of them they way they once did in their younger days (with you!). It's definitely worth a try. I've seen women go from raging & screaming to calm and smiling as they rock their little ones in their arms. https://www.amazon.com/DH-Beacon-Activities-Communication-Interaction/dp/B073294DCH/ref=as_sl_pc_qf_sp_asin_til?tag=agin0c-20&linkCode=w00&linkId=f20ec641204b12201bf2248501111c96&creativeASIN=B073294DCH

Elizabeth, time to bring out the "therapeutic fibs". Make up something you feel that your Mom would believe, such as "there is issue with the water on your street, can't return until it is fixed".

It's really tough when they are stuck on something. Sometimes you have to leave the room yes. Or you can gently redirect her thoughts to something else. If that doesn't work, leave the room.