I have used the "hum" technique to produce the growl since 1956. Both the growl and the "flutter tongue" were very popular techniques used by sax players in Rock & Roll and R&B music which was hitting the airwaves in the late 50s. Those of us who joined the local school band and liked the new music were asking the band director what this new "gritty or raspy" sound was that we were hearing on the Top 40 hits featuring Lee Allen and Grady Gaines on tenor and Earl Bostic on alto. This effect was used before the 50s and is still very popular in many forms of music. The 1985 hit Rockin' at Midnight by The Honeydrippers features Keith Evans using a lot of this technique in his tenor solo.
There are other ways to achieve the growl effect from what I have read on the NET over the years but I have only used the method of humming while playing a note to get the growl tone.
I suggest that the best way to learn this technique is to start with the mouthpiece attached to the neck. If you use the entire horn it can be a problem at first due to the coordination involved. Play a note on the neck then start "humming" a note that is higher or lower than the pitch that is coming out of the neck. Some players hum in a falsetto range to get above the note that is produced by the neckpiece. Some players will sing/hum a lower note. For example, when I play a G above the staff on tenor I find that I usually hum the pitch that is close to D below the G. If you hum the same pitch that is coming out of the neckpiece or sax the effect will be cancelled.
At first you may feel that it is taking a lot of air to play and hum at the same time on the neckpiece. This is natural because you will probably open your throat and exhale too much air as a result of your efforts to hum and exhale at the same time. Easier said than done at first! In time you will learn to control the amount of hum and the coordination will become natural. Now put the horn together and see what happens.
Many of us use the growl in the middle and high range,
especially from high A above the staff to high F#. I have found that the most effective area for the growl is starting on 2nd space A of the staff and upward. Once you go below 2nd line G it becomes somewhat garbled.
I have read articles that suggest that you hum a 3rd above the note being produced on the horn. However, while on stage during a rockin' performance I can't hear the note that I am humming due to the stage volume. I have never thought about the "3rd above" concept so I can't comment on it however if it works for you that is all that matters. I hum in a range that is usually below the notes being played which works for me.
As you practice this effect ask for more advice from other local sax players and review articles on the internet. This is valuable because there is always more than one way to approach any effect.
Let The Good Times Roll - Shirley And Lee - Lee Allen - Tenor
Slippin' And Slidin' - Little Richard - Lee Allen - Tenor
Keep A Knockin' - Little Richard - Grady Gaines - Tenor
Super Freak - Rick James - Daniel Le'melle - Tenor
Unchain My Heart - Joe Cocker - Clarence Clemons - Tenor
The Heat Is On - Glen Frey - David Woodford - Tenor
Rockin' At Midnight - Honeydrippers - Keith Evans - Tenor
Harden My Heart - Quarterflash - Rindy Ross - Alto
Freeway Of Love - Aretha Franklin - Clarence Clemons - Tenor
Other links for the growl: