The cambium is not the only living part of a tree. The foliage is alive. The phloem is alive. And part of the sapwood is alive. Without living sapwood, the tree could not conduct water.
I'll have to do more research but I find it hard to believe that a tree can continue drawing water long after it dies, because if it could still draw water, then it could still photosynthesize, etc, etc. I would expect xylem cavitation shortly after death.