"Now, in most states, your employer could fire you for engaging in immoral behavior off the job, "
Not unless you have them sign a morals contract first. Of course you could make up some other reason to fire them but if you fire someone for morals they can not only claim unemployment insurance but could also sue you, assuming it was worthwhile.
Almost all (if not all) American states are "Employment-at-will" States. That means you can fire employees for a good reason, bad reason (except illegal reasons, such as firings based on race) or no reason at all. It is perfectly legal to fire somebody for engaging in what you consider to be immoral reasons, even if the conduct occurred outside of the workplace. Whether it's a smart employment practice is another question, of course.
However, these folks are all government employees and their employment is probably governed by a collective bargaining agreement that lays out under what circumstances they can be fired. I would be surprised if there was a generic morals clause in their contract.