lots of options
try playing a vintage sax in a music shop – lots of old blues players will tell you this (so all you guys saying the choice of sax doesn’t make a difference to how your playing sounds need to think again).
however i wouldn’t advise a learner to get a vintage sax.

one way is to store some reeds in water with some mouth wash, when you aren’t playing the reed – this does make a difference to the reed playing sound.

another option is to change your mouthpiece/reed set up.

Also it helps if you are good at bending notes and doing scoups and all that sort of stuff to get bluesy. Some condider it as playing with a different accent, just like you would put on a foreign accent when talking, it can also be playing more flattish or sharpish

lots of stuff to try out.

you don’t need a tenor to play the blues…