I had a NYS driver's license until the end of May, 2009 but lived and worked in Tennessee for that entire period. I moved to Ohio the end of May and got an Ohio driver's license and a permanent Ohio address (living with my parents). I work in Ohio since that time. What, if any, tax forms do I have to do for 2009 for NYS? Am I considered a full-time Ohio resident for 2009?

You were a part-year rsident of TN (from Jan - May), and a part-year resident of OH (from May - Dec). Since you neither lived nor worked in NY State at any time during 2009, you have no obligation to file anything with NY State. The fact that you carried a NY drivers license until May is immaterial - what's important is where you lived during this time.