I have been playing the saxophone for about 4 years, but primarily in the style of classical music. I've always been interested in jazz saxophone, and want to learn how to actually play it. How should I go about learning the theory and technique necessary to decently play a gig with a combo for instance?
You probably have the chops to play from your classical training. But Jazz is highly improvisational and the only way to learn improv is to jump in the deep end and swim. "Jazz theory" for what it's worth derives a lot form classical music theory so what you know will take you far. There are different styles of Jazz each with their unique feeling. If your sight reading skills are good you would do well in a big band as that is closer to the orchestral setting and solos tend to be short with an emphasis on variation of the main theme. In a bebop group you would be expected to completely rewrite the tune in real time (so to speak). The danger in relying on sight reading for that type of music is that your main score will be a real book and there is not a lot of information there. Like cliff notes, you get the terrain of the tune but you need to KNOW the tune to get any value out of that. The only way to know each style is to immerse yourself in it. Listen to these tunes and get them in your head. Listen to Duke Ellington, Miles Davis, Thelonious Monk, Wes Montgomery, John Coltrane, Dexter Gordon, Dizzy Gillespie, Charlie Parker, Charles Mingus, A. C. Jobim ... etc, etc. As for show tunes, listen to original versions and also covers by the players mentioned above. When you hear something you like, transcribe it. The theory of chords and progressions will be of value but will not likely help you produce "jazzy" sounding music by a formula, the real learning comes from liking the style and imitation. After some time you will begin to invent your own unique musical language for improv. Then you are playing jazz.
Start transcribing solos of famous jazz artists on famous jazz standards. This will get you listening to the music, which will develop your ear for jazz and your jazz ideas. Transcribing will also help you learn songs that are likely to be played at gigs. It will improve your ability to quickly translate a musical idea into actual notes and then your instrument, which is crucial for improvising. Finally, it will use your existing theory and technical knowledge as a springboard for how to construct a solo. Creating a melodic solo requires a ready repertoire of musical ideas, and there are no better teachers than famous jazz musicians like Charlie Parker, John Coltrane, etc.
I like to think about transcribing as a process that culminates with playing along with the recording. Transcribing is an outstanding way to train one's ears and ideas, but it can also be a way to exercise one's technique by mimicking the recording.
It might sound like I'm advocating for you to use transcriptions as a means for learning to mimic someone else's style--with no room for original thought. That's not the case. Even after just a little transcribing, I think you'll find that your own ideas start to flow much more freely and plentifully.