With a score of 623, your rank might be approximately between 2788-3956 . This is an expected rank range: https://medicine.careers360.com/articles/neet-result-marks-vs-rank . Now regarding the AIIMS Colleges that you may have a shot at, you haven't mentioned your Category. Nevertheless, here you go:

• SVMC, Tirupathi
• Kurnool Medical College
• Andhra Medical College

Having said that, this is based on 2019 AIIMS data and cut offs change every year. Cut offs change every year depending on various factors such as no.of Candidates appearing for the exam, no.of Candidates qualified, top NEET score and difficulty of the exam. For a complete list of colleges that you may have a shot at, please use our College predictor: https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer