How can I find an elder care home for my wife (who has dementia) in a state beyond New York where care costs range from $6,000/month upward?

I’m not certain what you mean by getting an apartment and putting her in “home care”.

Facilities nationwide all charge thousands a month for their care. NYC is not unique. In home care is pretty much the same as well. Have you considered applying for Medicaid?

In the metro area of Minneapolis $6K is average. You'd need to live in the rural areas to get a better price, but then you're not by conveniences, art/culture, public transportation, better quality medical care, good wifi, family...poorer counties have fewer services. It will be a trade-off no matter what.

In Colorado, I was paying $4500 a month for a studio apartment in Assisted Living with medicine administration only for my mom, with yearly increases. I pay $6400 a month for Memory Care at the same AL, which doesn't include Depends or phone or other extras. That's for a hotel sized bedroom and large bath. Skilled Nursing is $8500 a month for a shared room, and $10,500 a month for private, up to $11,500 in the upper end SNF. I think these costs are normal wherever you go, unfortunately. If you can swing it, do it.......in home care becomes too much of a burden for most of us, especially when dealing with dementia.

what I meant by my letter was that I would get myself a small rental in a city at which I could buy assisted living care for my wife at a price that was lower than that of NYC.My living quarters would be near to my wife's living place. I would then be able to visit her daily and help her deal with our enforced separation re her personal care.
When I had to take her to a rehab center after her release from surgery and recovery, I am convinced that her intellect took a 20-30% drop from her enforced separation from me each time I visited and then had to leave. And, I must confess, I felt terrible having to leave her each evening when I left her rehab home. So, what I seek is a less expensive cost at assisted living for my wife and a place where I can get a regular tiny apartment for myself. I am a writer of stories, books, etc. so I could stay in any apt. that would accommodate my computer, my bed, a desk, minimal kitchen facility (e.g.g and my own private bathroom). But it appears almost as if there is some sort of cartel in this country that pushes every establishment's charge for housing the elderly deficient into a cost that is upwards of $5,000/mo and then adds on $1,000 increments/mo for every additional service they perform. Mental care (for dementia) adds about $1,700 per month.
The relative uniformity, across the U.S., of the fees for the housing of the elderly or mentally troubled in assisted living, seems to fly in the face of reason. The cost of the buildings, the level of pay for help, etc. is probably very different from New York City to a place in middle America, the south, and other possible living areas. Why should the fees charged for the AL services being rendered not vary accordingly?
My thanks to those who sent sample numbers re fees for care.

Well, have you given any consideration to both of you moving into AL? We are placing my step father & wife into one here in Florida. 700 sq ft, 1 bed, 1 1/2 bath, nice kitchen. It is beautiful place, has a computer room, theater room, gym, therapy pool, bistro and more! 700 sq ft is larger than the apartment I had when I lived in NYC. It will cost them 6K per month, includes the extra person charge, med distribution, for both, bathing for him. If they go into memory care (her), it will include all the above and more for around the same $$. The reason they are not in MC now is because he needs to go to dialysis 3 times a week and it would just add to the confusion, so far he can manage her, she is not a roamer, just the opposite she won't go or do anything unless he gives her permission.

I live in NJ, southern part.

Have you looked into a facility where it has independent living and Assisted living and LTC too. You could live in the IL. You get meals and other things. Your wife goes to the AL or LTC. You will be close enough to visit.

I would check out Medicaid. As a Community Spouse, they will split your income. Hers will need to be spent down. Once that is done, Medicaid can be applied for. To do this though, she will need to be in LTC.

I suggest you call your local Office of Aging. Ask if there is someone there who can help you with options.

Here’s some small town upstate NY prices. Moms Independent Living apartment in the Syracuse NY area was $3400. Some couples lived there as well. I augmented this with private caregivers for meds and help with bathing a few hours a day for about $600 a month. When I checked (3 years ago) into assisted living it was about $4000 plus add-ons a month, Memory care was about $5000. Her NH was about $11K a month. I second the suggestion already given to look into a place where you could both move, like Independent Living with extra caregivers, or assisted living. That sounds like it might be an easier solution for you both.