WBN, one of your issues is that you are facing not only dementia but other health issues. Managing the first is hard enough but managing the others around the first is harder.
In terms of dementia the solution presumably is to identify the extent of the disability and work out what support is needed. The first issue then is to have the person accept the problem and agree to support. This is really difficult but it appears you are some way there. Then it could become some form of options analysis, which should include the money, your time and support from social services which might be available.
I provided something like this for DB, though he failed to respond, which included a care home (too early), carer support in her own home (would she like someone in her own home and how sustainable would it be into the longer term), and different forms of supported housing (here 24 hour warden support was essential, however without being sure that she would settle/manage buying a flat was a financial risk).
I would be tempted to tell her you were looking at options and in order to do so you need to be absolutely sure of her finance....could she hand over the paperwork.
Then ask everyone you know who goes to a lot of places. Professionals such as SS and the memory service wont be able to make recommendations but could suggest places you might visit. Try the GP, the vicar/priest, others such as the physio if she has one. A British Gas engineer was able to tell me that he regularly visited the place my mother now lives in, and that it was lovely and the staff very kind.
In terms of distance I don't have a problem with 150 miles each way if it allows my mother to retain her social networks. However my mother's general health is really good and we were never that close, so she will not expect to see me that often. I can time my visits to avoid traffic, and often combine it with something else. Not least I have joined the National Trust and am planning to visit everywhere on route sometime in the next year.
However if my support were needed for other health issues, including visits to doctors and hospitals and also making sure my mother was in for routine visits from community professionals, as well as predictable emergencies, I would want her to be closer. Also because my mother's health is so good, her prognosis is long; when it was all at its worst I felt she would outlive me. (I understand completely what Lavender is saying.) Your mother has a much shorter prognosis. DC are also teenagers and so can be left.
It is great your mother is talking about a move. You should encourage her to move "early". The more skills she still has in place the better she will be able to settle, adapt and make new friends, and so the better the quality of life going forward.
Best wishes to all....