I do not tell my employer everything, that is a scenario to demonstrate that you doing a second job violates basic trust, or even the contract. That is different than you looking for a new job, which should always be at the back of any manager's mind.

great, then go on social media and bad mouth your employer and see if HR bats an eye, they should be only concerned about your output right.

Why would it vary by state, if it isn't a legal requirement in any of them.

It is purely on the employer's end, and I cannot find a legal restriction against it in any state. So it would only vary by employer.