If you're sure that you want to create a will trust, then you do need to become tenants in common (rather than joint tenants, as I assume that you are now).
As joint tenants, neither of you owns any individual share of your house; instead, it's your 'partnership' that owns the whole of the house. When one of you dies, the surviving partner will automatically own the whole of house. (Nothing written into a will can change that because a person can't bequeath what he or she doesn't actually own, as such). So any provision in a will that sought to pass half of the ownership of the property to a trust would simply be invalid.
As tenants in common though, each of you would own half of the property. As such, each of you could choose what should happen to that half of the property upon your death. (You could, for example, leave it to Battersea Dogs' Home if you so chose). You can therefore write a will which places your half of the property into a trust upon your death. Then if the surviving partner needed to go into a care home, that half of the property couldn't be touched to pay for care costs, meaning that it could later be passed on through the trust to your children (or whoever else you've chosen to benefit from the trust once both partners have passed away).
A relevant link:
https://www.which.co.uk/money/wills-and-probate/passing-on-your-money/will-trusts-and-lifetime-trusts-aqmf66w4nu5w