This unfortunately doesn't solve the stickiness problem of (say) a shopping cart: you don't have a way to back-propagate adding an item to a cart to the old continuations. You could can have them just keep pointers to state, not the state itself, but then you end up needing to maintain that too, which is something continuations set out to avoid.
Don't get me wrong, I'd love for this stuff to work magically well and let us be rid of the pain that is application programming over a stateless protocol. Maybe continuations have their place in the infrastructure of the Non Sucking Web. But they're not all that's required.