Medicine without doctors? State laws are changing who treats patients.

Arlene Wright is a doctor, but she doesn’t want people to call her one.

“I usually tell my patients ‘your majesty’ or ‘your highness’ is fine,” she said.

Wright has been a nurse for more than 20 years in Fort Myers, Fla. She began working in hospitals as a teenage candy striper in Upstate New York, progressing through an associate’s degree in nursing, then a bachelor’s, then a master’s, then finally a doctorate of nursing practice in 2013.

Wright has always told patients she’s a nurse practitioner, she says. She doesn’t flaunt her doctorate or try to mislead patients into thinking she has an MD.

Still, when Florida lawmakers began considering a bill that...