I was chasing a rabbit across a desert trying to get food.  When I finally killed it with my Fire Aspect II sword (full enchants: diamond sword, named, has Sharpness V Sweeping Edge III Looting III Fire Aspect II Unbreaking III Mending), I got raw rabbit.  However, when I kill rabbits not running away from me, it usually (but not always) drops the cooked meat.  I did some more tests and I think if the rabbit is moving or about to move, it drops raw meat.