Is it enough for me to show that a property owner knew or should have known that the area was generally dangerous in order for me to win my slip or trip and fall case in New York?

No; in order for you to survive a motion for summary judgment in a slip or trip and fall case in New York (assuming that you can't prove that the defendant either created the defective condition or had actual knowledge of the defect before the accident), you will need to prove that the defendant should have known about that specific dangerous condition. For additional information on this topic, please see "Why 'General Awareness' of a Defect Isn't Enough to Win a Slip & Fall Case in New York."