I absolutely agree with the posts above. Familiarity of your instrument is essential.
I am one of those blessed with good time from the get go, but there's always room for improvement. Being overly critical of something so important has led me to some effective teaching tools and some exercises that help me get my time stronger every day.
I find that my students who have trouble keeping time have other technical issues.
First, most players that are fresh in the jazz-world don't swing properly. I know this might open a can of worms, but hear me out. They swing with their fingers. What happens is that they are trying to swing so hard, and their finger movement is uneven - their brain can't process all of the information. Your fingers get wrapped up in each other and you fall out of time. Anything you play needs to be played straight and practiced in several different articulation patterns. The articulation gives it the swing. Slow scales (all keys), with a metronome and very straight. This will get you familiar with the horn, and boost your technique. Once you feel comfortable with all of this, then articulation with make you swing.
"Jazz articulation" It's counter-intuitive to what we are taught from many books (especially beginner band or learn-to-play). I have my students take their major scales and have them articulate, in eighth notes - tah, tee-aah, tee-ahh, etc. You tongue the first note and then every off-beat after that. This forces you to swing. One important way, at least to me, is practising this with the metronome beating on 2 and 4. It simulates a traditional hihat that you can lock into when you're on the stage. When you get the appropriate articulation happening, you'll stay in time and you can focus more on where you are in the form.
When I was in college, albeit briefly, one of my profs (Adrean Farrugia - killer pianist, check him out) told me to go buy a ride cymbal. I did. Just focusing on the quarter note (a la Jimmy Cobb) changed my perspective on time (and focus). I usually do this for 15-20 minutes a day. Something relaxed for about 10 minutes, and something that pushes my technique (physical and aural). I have found this to be a very worthwhile exercise.
One more thing that I get (force. lol) my students to do is sing. Any exercise, any leadsheet, any transcription - it doesn't matter. If the saxophone is an extension of our voice, we have to be good singers. I don't mean Ellas, or Sinatras, but being able to reproduce rhythm and pitch within the saxophones' spectrum. If you sing, you can internalize what you need to know without the piece of plumbing in front of your face. No distractions! Your time is naked! Take Lennie Niehaus' Basic Jazz Conception Bk. 1, sing the first exercise. I'll bet that your time gets better.
I hope that this helps.
If you have any questions, PM me.
Cheers!
Rich