Assuming you moved to AZ during the tax year, you'll be completing a part-year resident AZ tax return (AZ Form 140PY). On that form, AZ wants you to include all AZ-source business income (such as rental income) you earned both before and after you became an AZ resident. So no, you do not pro-rate for AZ. (See Business Income on page 8 of this reference: https://www.azdor.gov/Portals/0/ADOR-forms/TY2016/10100/140PYBooklet.pdf)
You'll also have to do a part-year resident SC return, on which you must include the portion of your AZ rental income that you earned while an SC resident. But you'll be able to take a credit on your SC return for the tax you paid to AZ on that portion of the income, so you won't be double-taxed.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
