You very well may want to keep making it so the texture is cakey. Some people like it that way.
Whether your cheesecake is cakey, or light-and-airy, you can always cut it with unflavored dental floss, and a cake server to remove the slice. It's the only way I've cut my homemade cheesecakes. This way, you don't need to worry about cheesecake sticking to the knife.
If you really don't care for cakey cheesecake (I do believe that is authentic NY style), follow these mixing directions for a tall and creamy cheesecake. The key is to mix for a longer period of time. NY Style cheesecake calls for mixing only just until ingredients are incorporated.
Working with a stand mixer, preferably fitted with the paddle attachment, or with a hand mixer in a large bowl, beat the cream cheese at medium speed until it is soft and creamy, about 4 minutes. With the mixer running, add the sugar and salt and continue to beat for another 4 minutes or so, until the cream cheese is light. Beat in the vanilla. Add the eggs, one by one, beating for 1 full minute after each addition — you want a well-aerated batter. Reduce the mixer speed to low and stir in the sour cream and/or heavy cream.
From the recipe for Tall and Creamy Cheesecake. Recipe from Dorie Greenspan. Adapted by Emily Weinstein. Listed on the NY Times Cooking website.