You haven't mentioned your Category. With a score of 560, your rank might be approximately between 19,000-22,000. This is an expected rank range: https://medicine.careers360.com/articles/neet-result-marks-vs-rank . Now regarding the Colleges that you may have a shot at in other States, you cannot participate in their Government Quota seats Counselling and so you have to try under Management Quota seats. These are usually more than 40 lakhs for the entire duration of the course. But you can try for Colleges in Karnataka and UP like:

• Era Medical College
• St. John's Medical College
• Kempegowda Institute of Medical Sciences
• M S Ramaiah

Having said that, this is based on 2019 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