Rusbun my parents faced exactly the dilemma you face over 40 years ago;wanting to move, in their case from a big family home to a bungalow - in the Epsom area, only to find that because they are so few they command higher prices than they are worth.
In their case they ended moving away from the area to the Sussex coast where they were able to buy a good quality bungalow with a nice garden and lived there very happily for 25 years
Not much help for you, although moving elsewhere is another option.
We have an interest only mortgage, which we took out 2 years ago when we had an extension build. and when we sell, either to down size or go into care, the house will be sold, the mortgage will be repaid and what is left will be what is available for another house or to fund a place in care. It is quite straight forward and such a mortgage has no repercussions related to care.
The other thing to do is take out equity release. Here you borrow money against the value in your house, but instead of paying the interest on the loan, you let the interest mount up, compound interest and it is paid with the capital when the house is sold.
I have just noticed that you are both in your early 60s, so steer clear of equity release, mortgage interest unpaid and compounded over possibly 20 years or more could completely wipe out your capital, leaving nothing for your family to inherit.
My personal preference would be for a Retirement Interest Only Mortgage (RIO), which is what we have done, but you do need to think very carefully as to whether you can afford the interest rate. We have a 5 year fixed rate, but the cost of it could double when the 5 years expire, which we will be able to afford, but may sell before then.