The question isn’t so much whether they allow an exception, and more who the the law allows to determine if the exception applies.
Ohio’s recent amendment leaves it with the mother’s regular doctor:
However, abortion may be prohibited after fetal viability. But in no case may such an abortion be prohibited if in the professional judgment of the pregnant patient’s treating physician it is necessary to protect the pregnant patient’s life or health.
Contrast with Texas laws, where that determination is subject to judicial review, and the physician can be convicted of a felony if they can’t prove the mother was sufficiently endangered.
Ohio’s late-term exception is a legitimate exception. Texas’s “exception” is non-existent.