Regarding playing, I like your ideas, and the approach you take. Where you need some work is intonation

Really it's a lack of breath support, because you are trying to make your lips (embouchure) do what your breath should do. You're flat in the bottom register and sharp in the top, try to not move your embouchure at all and really work on breath support. Do some long tones, maybe with a tuner, or with a drone tone to get the intonation locked in.
Regarding the recording, if you have $100 (maybe less), invest in a decent mic. A MXL condenser mic would help a lot. Stand about 2-3 feet back from the mic, and have it pointed at the middle of the horn, right at the pinky table. Your recording interface is fine.
Regarding mixing, the sax part needs to be more up front, i.e. mixed a little louder, but more importantly it needs to occupy the same sonic space as the backing track. Add some reverb and more importantly EQ. Also some compression, though that's harder to get right.
I recommend checking out the "Recording Revolution" Youtube channel, Graham Cochrane puts out a video a week (or more) on recording and mixing techniques. Of course he is trying to upsell his expensive courses, but his videos are very good. He has a bunch of videos about how to have a complete recording setup for under $350 - that includes interface, mic and headphones. Pay attention to what he does with vocal recordings, because for this music your sax is the vocal
Keep at it, you are definitely on a good path! Good luck!