I would recommend to listen to some players that are into the "Chicago BluesSax Style". IMO this style have less in common with jazz! And the Chicago Blues Sax Style is also much about the electric guitar and sax.
J.T. Brown (Elmore James ....), A.C Reed (Earl Hooker ...... Rolling Stones), Eddie Shaw (Muddy Waters, Howlin Wolf ....).
Another saxplayer that is important to the Chicago Blues and Chicago sound is Gene Barge (Little Milton, Koko Tayor, ..... Rolling Stones) and also Harold Ashby (went on to be soloist with Duke Ellington) did lots of classic Chicago Blues from Otis Rush to Willie Dixon/Memphis Slim. Abb Locke on Koko Taylor's "I Got What It takes" and "Earthshaker" (Alligator Records) is also a great sample of Chicago Blues Sax. There are other good saxplayers pre Muddy Waters and Howlin Wolf era.
The Chicago Blues Sax is often about a single sax in a band.