I've recently joined a big band. I play second alto saxophone. I do well enough when playing the slower pieces. But I struggle to read the quicker swing pieces. What's the best technique to play faster swing rhythms?
ggcg's comment is correct. practice. Specifically practice sight reading.
Sight reading is just learning to recognize patterns you've seen before. The only way to do this is see them a lot.
Maybe ask the musical director for some parts to take home each week (pieces you aren't already playing) and read them at home. Don't read them two days in a row. Don't practice them. Meaning, don't stop to correct mistakes. Only play them at most twice a day. Play them as slowly as you can without mistakes. You will get faster in time.
There is also a skill of being able to quickly glance at a part and identifying key parts. Key changes, repeat signs, ds al codas , runs or other possibly technically difficult parts, etc that may cause trouble. Being able to spot this and do a quick mental assessment before the leader counts you in gives you an edge up. Practice doing this as well. At first maybe take longer doing this so you get better at it and then slowly reduce the amount of time you spend on this pre-playing "cheating" as you get better at it.
But yeah...practice practice practice.
First thing I do, given a moment, is to highlight the repeats, DCs,DSs, codas. And key changes. Then at least I can easily follow the geography of the piece.
Looking ahead, as ever, and count like hell ! Keep a foot tapping, head nodding, whatever works best. For me, having an idea of the chord structure helps, as it allows clues as to how the tune may go melodically. Main notes are often chord tones.
These other answers are good.
I just want to touch on the "swing" part. A problem you might be having is that you're trying to focus too much on swinging.
Firstly, you should not be swinging with your fingers. If you're swinging by moving your fingers as jagged speeds to get to the different notes, then you're going to seriously inhibit your ability to play fast. Nearly all swinging should just be done with your articulation.
Secondly, you should start integrating swing into your regular practice sessions. Run your scales and modes full range with a metronome at half speed. Treat this metronome as the hi-hat hitting on beats 2 and 4. Slur every other note (think "ta too-a too-a too-a...") but keep your fingers consistent. Eventually, this will become second-nature and you won't have to think about swinging.