Is it actually possible to express the entire universe as a single time-independent equation?

Yes.

If your original question, "what determines how much the universe changes from state to state?", is meant to refer to spacelike "states", then the answer (which requires only general relativity) is the geometry of spacetime. But the "states" in the Wheeler--DeWitt equation are spacetimes, so in that context "the universe" differs "from state to state", but it doesn't "change."