You love the saxophone and you picked it up and began to learn to play. You’ve improved greatly to some extent; you can play many songs, give great interludes to songs.
You can also manage to solo once in while but you are not-satisfied because most times it’s just as if you are playing out of context.
It seems like the sound you produce is odd and does not even rhyme with the flow of the band.
Is this you? Then you need to learn a new musical instrument. Your aim of learning to play this musical instrument is not to become a pro at it but to make you understand a concept in music that I call “Chord changes” or “progression”.
If you want to learn to solo effortlessly with an endless flow of solo lines inspiration, you need to understand the chord changes of the song in context or the progression of the song in context.
I’d recommend that you pick up the bass guitar as your second musical instrument. It’s easy to learn and you also do not have to do too many exercises(at least for this purpose).