Hey Liz, great job overall on this recording. You've got a strong voice, good solid tone and some nice musical ideas starting to form. I second the comment that having a good teacher is important, although in a pinch this kind of remote "tech support" can help. Before I continue, I want you to understand that any criticism here is not intended in any way to make you feel bad or discouraged... quite the contrary. Any advice or suggestions are intended to identify areas that could be improved to take your playing to the next level. All players have a "next level" available to them, even the greatest players in the world. This is what makes playing the horn so fun -- there's always something to work on.
SO... getting into specifics.
The first thing to think about when soloing is how time and note choices fit together. To execute a great solo is to combine these elements in a strong, interesting and comprehensible way. Our ears naturally gravitate toward strong and weak beats and notes. If you listen to something simple like a major scale, these strong and weak notes are very apparent. In a C major scale, the B pulling to the C is the strongest, F to G is second strongest, D to E is third and so on. Every chord is surrounded by these leading tones a half step away in either direction. So, a C major chord can be lead to by 6 different notes, each with it's own character. Steve Neff (Nefertiti on here) has a series of exercises he wrote that I highly recommend that exploit all of these possbilities.
So, now you have a way of getting to the chord, the next is knowing WHEN to get to the chord. Timing of phrases is super important, because you need to be aware of strong and weak beats, just like strong and weak notes. A strong beat is a down beat and the strongest of all is the first beat of the measure. Starting a phrase on a strong beat is something almost all inexperienced jazz players do, because it seems logical. It's how we practice scales, it's how etudes start, etc. The problem is that it makes a weak phrase in jazz. If you listen to the greats, especially how they start phrases, you almost NEVER hear them start anything new on a strong beat, unless it is for a very specific effect. Once you realize this, everything starts falling into place. And, if you combine the strong/weak NOTE choices with phrasing this way, your playing will take a quantum leap forward. A good technique to work on this is to start major scales on the AND of four, with the leading tone. So, B, C, D... etc. You will hear how strong and weak notes/beats line up, except at the very end. You can add in the flat 7 to make the pattern work better, so in C the last three notes would be Bb, B, C. You can add other leading notes in as long as you keep the weak/strong pattern going.
When you are playing, I hear a lot of sustained notes that cross bar lines and chord changes. When you pick a note to hang out on, it can be really cool if it is a note that resolves to the next chord. But, in the blues, there are not a whole lot of these that work, and doing so is kind of cliche. If you want to set up something that crosses chord changes, find a simple phrase that you can repeat across different chord changes. You hear great blues guitarists do this all the time. Experiment with these 3 and 4 note patterns and you'll find some cool things that work in any blues key, and even on standards sometimes. In terms of general phrasing, think in terms of 8 bar segments. Your phrasing tends to consistently be 4 bars long, which is fine, but longer ideas will by force be more developed.
In terms of saxophone stuff, you have a good base tone. One thing you should look at is how much diaphragm support you are giving your sound. You want to make sure that you're breathing from low in your stomach as opposed to the upper part of your lungs. The difference is, when you breathe in, your tummy should rise, instead of your shoulders. Then, when you push the air out, the muscles of your stomach are flexed, like you are doing a sit up. There's no "huuuuuhhhhh" sensation, it's more like deflating a tire "tssssssssssssssssss!!" where there is a lot of pressure and control. Another thing I hear in your tone is heavy emphasis on the fundamental. What this means is that of the harmonic spectrum that composes a note (Google overtone series), you are unconsciously focusing on the lowest notes in the "stack" -- which makes your tone more compact and less vibrant. This will also keep you from executing altissimo because that depends on having the ability to "activate" those higher partials of a note. A way to begin developing a richer, more resonant sound is to practice playing notes without the octave key. Try playing a C scale all the way up to high C without using the octave key at all. Once you master this, try just playing random notes without it. Next, try playing a low Bb, then make it an octave up, and then push it higher to F. Evetually, you'll be able to play "bugle calls" with nothing but lip and diaphragm. A teacher would be especially helpful when you get into this. From that point, your tone will be way better, and altissimo will be a piece of cake.
One other saxophone-specific thing... your articulation is focusing on strong beats. You should practice breath accenting the off-beat notes, as well as understanding syncopated articulations like Cannonball used (lots of groups of three). Listening is key here. Even though he played tenor, check out Dexter Gordon. He has some of the most eloquent and accessible phrasing of anyone out there.
Lastly, I wrote an article on here a while ago about the "swing limit" -- and you definitely want to check it out. You're doing a great job at this tempo, and you're not playing a bunch of extraneous notes, but the desire to do so is almost inescapable for sax players, especially those with good technique. (guilty as charged here!). A rule of operation is to be very selective about what you play. This includes notes, rhythms and inclusion of space between phrases. A great player like Miles Davis or Paul Desmond was almost known more for what they didn't play, as opposed to what they did play. They both used space and simplicity to an amazing degree. If you say something great, give time for it to sink in. This is another thing you'll find with all great players.
I know this is a lot to digest, but if you can work through it, these are all the keys you need to get on the road with basic jazz improv. From there, it's learning jazz harmony, rhythmic expansion, and how to get around on your horn.
Great job, and keep up the good work.