Edit: I am taking a break now and will return later this afternoon/evening to respond to any comments and answer any questions. Thank you everyone for a great and engaged AMA so far.
I called the national passport line for advice about this when renewing, and they said to mail in the correction form the other commenter linked along with the renewal form. I also included a letter explaining that I wanted the passport renewed, but that it had my wrong birth state, and I wanted that corrected along with the renewal.
I think I included my birth certificate showing where I had actually been born, also, but I don't really remember. I'd call the State Department national hotline to see what they suggest: https://travel.state.gov/content/travel/en/contact-us/passpo...