Can we use one medication administration policy across several states?

You should not. In assisted living, medication administration is frequently delegated to unlicensed staff under state-specific delegation rules, and what may be delegated, who may supervise it and what must be documented differ substantially between states.

A multi-state operator running one national policy has by definition written a policy that does not match at least some of its states. When a medication error occurs there, the claim arrives with a regulatory violation attached, which is materially harder to defend and more likely to attract a heightened-conduct count.

Write the policy state by state and make the training record reflect which version each staff member was trained on. That record is what a surveyor asks for, what defense counsel needs, and what an underwriter reads as evidence of management quality.