After you have had your sax checked out, this exercise I used to use with my students might help.
Play low G
forte and then quickly slur down to low C and hold it as long as you can. Shape your throat as if singing "AHH" on the lowest note you can hit and blow warm air. Resist the tendency to relax the embouchure for the lowest notes for now. Later when you work on "subtone" that will change.
After you can sustain low C as a long tone at
forte, repeat the exercise at
mf, and then
mp. Once you have learned the "taste of the note" (embouchure, air, and voicing) and can produce the note consistently, then try starting on the note itself.
Each size of saxophone has its own unique feel when playing the lowest notes in the range. Because you can do it on alto doesn't mean you can automatically do the same on tenor. For some reason the bari seems to be the easiest. When scientists use an "artificial" embouchure to play a saxophone for acoustic studies, the machine can be set to do nearly everything a human player can do---except play the lowest notes where the fundamental is weaker that then next two or three overtones. It takes the nuanced control of a human oral cavity to pull this off---something science has yet been able to replicate. Tonguing and fingering "Giant Steps" is relatively easy by comparison.
