Does it sound the same if you start on the note and if you change to it? I find I sometimes struggle to get the low notes immediately and ‘fall’ into them. I have been playing about 6 weeks now and it’s improving all the time but still a way to go obviously.
Congratulations! You are already playing "multiphonics" something that takes others years of practice for most players. Seriously, I am familiar with exactly what is happening. The shape inside your oral cavity and the speed and direction of the airstream isn't locked into the low C fundamental, or the 1st harmonic the C an octave higher so your sax is "confused" as to which one to play.
Here is one method I use with my students. Sing the note low C on an "AHHH" using lots of warm air. This is how the oral cavity, throat, and air stream should feel when playing that note. Next, start on low G and quickly slur down to low C and hold the note as long as you can on one breath. Remember to keep the throat open and use lots of air, blowing more air as you go down.
Once low C becomes stable, try coming in on low C. Start the tone with either "tah" or "daw" not "tee" and remember to keep the back of the tongue down and use lots of air. When this can be done consistently, then start on low C and purposefully make it jump an octave by changing the shape of your tongue from "AHH" to "EEE" and blow faster air. This is how we play harmonics also called overtones on the sax.
Once you learn the "taste" of each octave when playing C you are now in charge with which one sounds at any given time. You might also want to revisit playing the multiphonic by setting your air and voicing somewhere inbetween.
If this method doesn't seem to be working at first, try checking the pitch of the mouthpiece and neck (crook). On alto it should be an Ab Concert, and on tenor an E Concert. If the pitch is higher, the embouchure may be too tight which makes the low notes harder to produce and control.