This is a very interesting question. What makes it fascinating is the fact that for almost all saxophones the fundamental of the lower notes on the instrument is weaker than the first 2 or 3 overtones. See
UNSW Saxophone Spectrum. This causes the lower note to want to jump to the higher octave---not the other way around as was described. It may be an octave vent position problem, but the problem with this analysis is that the body octave vent is usually set at the ideal position for the note F. Your E is only a half step away from this "ideal" position so the octave vent "not working" for this note would be unusual.
Something you might try.
Play E2 and Eb2 as overtones voicing the notes in this octave without using the octave key. Do this at
f and
ff and see what happens. Then play them at the same volume starting without the octave key and adding it while holding the note.
Another part of the puzzle may be the input pitch into the saxophone. Play just the neck and mouthpiece as you normally play and see what pitch is produced. This also may provide a clue as to what is happening acoustically. If you have pushed a pipe cleaner through the body octave vent and it is clear, I would be very surprised if there is a mechanical reason the sax is acting in this manner.
If you discover you sax just plays all of its lowest notes this effortlessly---how much would you sell it to me for? ;}