There are a few acoustic reasons for the D
1 and D
2 on the saxophone to behave the way they typically do. The sharpness of D2 is in part caused by the compromise position of the body octave vent. Ideally there would be 12 octave vents for each note of the chromatic scale, but this would be a mechanical nightmare so it has been reduced to just two. For the notes D
2 to G#
2 the body octave vent opens, and for the notes A
2 and above the neck octave vent opens.
On most saxes the body octave vent is set in the ideal position for the note F. The farther notes using that vent get from this note the sharper they become. D
2 and G#
2 are each 3 half steps from the ideal note's position so they are both sharp when played with the octave key. The neck octave vent is usually set at the ideal position for the note B. That causes both the A and C# which are two half steps away to be sharp when played with the octave key pressed.
The stuffiness of the D can be partly explained by the fact that it is one of the "under vented" notes on the saxophone. This is explained in detail in Curt Altarac's article on
Setting Key Heights. Simply stated when one plays a D on the saxophone the note vent's through the low C tone hole, but the next tone hole which is C# remains closed. The next closest tone hole for the D's soundwave is the low B. If D sounds like the soundwave is partly trapped inside the sax, it is because it really is. Low D can be vented by opening the C# key. That is impractical when playing D
2 because it makes a sharp note even sharper. Sometimes it is possible to open up the sound of the D by increasing the opening of the low C key, but care has to be taken to keep the pitch at a workable level when experimenting with this. A sharp D2 can be brought closer in tune by closing the low B key. This is only practical when the D2 is a long tone.
As has been suggested in other posts there are things the player can do as well, but dealing with D has always been a part of learning the geography of playing the saxophone.