Hello,
You can appear for the AEEE exam while studying in the Tamil Nadu State Board. The State Board syllabus for Physics, Chemistry, and Mathematics generally covers most of the topics required for AEEE. You do not need to switch to CBSE, but you may need to refer to additional reference books or practice previous year AEEE papers to cover any small gaps. Many State Board students prepare successfully using State Board textbooks along with coaching material or standard reference books for better preparation.
Hope this helps you.