Teacher from US looking for room in center max 600€

Hello! I'm a 53 year old English teacher from California. I'm moving to Madrid next Tuesday and am looking for a room to rent in the center. I'm flexible but would ideally like to live near a park. If possible, I would really like to get empadronamiento. I'm a friendly, social person who also likes to relax with a book or watch TV. I can pay 600 euros max. Thanks in advance!