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Yes and no.

An adjunct typically has a PhD or a masters degree combined with real-world experience.

Public school only require a bachelor's degree, but technically, teaching public school in most states also requires a teaching credential (especially with the No Child Left Behind law's "highly qualified standard), which is not part of most content (that is, non-teaching) programs. However, many districts are purportedly willing to hire a masters/PhD candidate and then back-fill coursework necessary for a credential (e.g., curriculum design, classroom management) as well as guide toward content certifications. These additional courses and certification exams are paid by the teach out-of-pocket, of course.

So, as with many things, YMMV.