Your fingers affect your tone on saxophone much more than you might think.
A master teacher named Henry Lindeman came up with a method of playing in the 1930s that places a great deal of emphasis on technique for a solid foundation in playing. Henry Lindeman taught Phil Sobel who taught Gary Meek, who I have learned a lot from.
Your air stream, tongue, and fingers all need to be aligned to get the tone you want.
And the lack of steadiness or movement that is too abrupt will disturb your sound on sax.
Check out Saxophone Tribe to learn much more and improve your technique (and sound) on saxophone.