I'm trying to imitate the Nasty Sax Sound that one hears in the bridge of Fats Domino's Ain't That A Shame.  So far I've vocalized the notes and then popped them into a MIDI Channel. The channel is set to play through one of Cakewalk's TTS1 channels which in turn is set to play back a Bari Sax.  The scheme works  and I am getting a kick out of doing this experiment.  My knowledge of Synthesizers is shallow so I'm not sure how to make the Sax  sound have some character.  At this point it produces a very very mechanical sound with no "Spitting attack", no dynamics, no vibrato etc.  It sounds just like one would expect a raw Midi to sound.  I don't want to encumber anyone but would welcome any suggestions as to how to make the sound closer to what a  real musician might create. Is there, for example, a plugin that does all the work or must I bite the bullet and learn synthesizer sound creation.   

Well ultimately you could use the Breath controller CC ( look it up) and set your joystick to that and try a few patches. I think some patchs in samplers will respond to velocity. You'll have to dig around your other synths for better samples. TTS-1 is famous for lo-fi sounds. I use it in my backing tracks sometimes.  But I would say the TTS-1 saxes are the worst sounding instruments in there. See what's in Dimension Pro or LE.  
I would try running it through a amp sim too. 
I need a better horn section myself and need to spend some time soloing patches. Read Greg Andertons Tips thread for a great way to preview sounds using the browser. 

Thanks for the info.  After penning my post I started looking at z3ta+ trying to find a PreSet.  With 1500 or more of them, there must be a  decent "Nasty Sounding Sax"   but the titles of the preset don't lend themselves to my way of interpreting things. I'll take a look at the other Soft Synths and will also check out Craig's tips. Thanks again Johnny!

You will need to get a sample set with a number articulations. A synth like Z3ta is not going to have that sort of sound. 

Getting a good sax sound with VSTis and MIDI is close to impossible. No matter how good you are at MIDI, you'll always end up with something that sounds like an accordion. It's much better to sample or record a real sax player.
That being said, counter intuitively, mixing some strings with a MIDI sax track can help.

It is difficult to get a good sound to replicate something perfectly (or even close) at times, and finding a preset when names often do not reflect the "sound" can be hit or miss as well (I have a truckload of Korg patches with patch names that are "meaningless" when searching). Dimension Pro has several presets that are not bad, and you can also adjust properties of the sound fairly easily within Dim Pro (there are some good top level videos for Dim Pro). I would suggest something that "gets the point across" rather than try to replicate it.
 
Cheap Trick remade that song with no saxophone, and it is easy to identify the song. (Actually, if the saxophone becomes a hurdle, you can do Cheap Trick's version instead )

If I'm correct you are talking about the rough sound a sax player gets from just blowing the crap out of the reed. I had an old Ensoniq EPS sampler that replicated that sound by cranking the vibrato speed as fast as it will go. Thus, when you turned the mod wheel you would get the "Nastiness".
I do agree that even now the technology still hasn't given us great sax or distorted electric guitar samples. I would look around locally or maybe find someone that would be willing to record the real thing thru the internet.
I do the bulk of my guitar overdubs now with Dropbox and Gobbler. My buds send me mp3s of the rough mix of their songs. I throw that into Sonar, do my guitar tracks, put the wav files in to Dropbox or Gobbler and they pick them up from there. They fly them in to their project and they are ready to go. I wait for the check in the mail. That's the best part.

A physical modelling type synth can make a convincing sax sounds (Yamaha's VL or Ex series, or Korg's MOSS synths for example). I think there is a VST synth, meledonia, but I never tried it.I believe there is a free player for it.

As I recall "Mr. Sax" uses a controller # for "growl".  Is that what you're after?
 
http://www.samplemodeling.om/en/products_saxes.php

Its called a growl as Jerry mentioned. Its achieved on a real instrument by humming as you play. The rough sound is caused by the discord of the human voice and the vibration of the reed. Its kind of fun to do for real.
If you can get a decent sax sound to begin with try adding what would be a low pitched male voice drone pitch and blend it in at low volume and see if that will do something for you. I have not tried this myself as I just play the real thing when its time. 
 
Good Luck.