I've played in my full share of big bands. Currently I am allowed to play with the Portland Jazz Orchestra, which I refer to as "sixteen people with music degrees . . . and me."
a) Reading
anything is better than reading nothing. Cello parts. Trombone parts. Real Book heads. Anything. It all helps.
b) Written music is a language in which several separate lines of content are presented simultaneously. Commencement of sound, duration of sound, articulation, volume and more are applied to each individual note, and to groups of notes. It's a lot to take in.
Phrased otherwise, the less you are taking in the better the chances are that you can nail (or at least reasonably approximate) the rest. For that reason I recommend hanging out with the Louis Bellson drum method,
Modern Rhythms In 4/4. You can play whatever notes you like -- drone, scale, arpeggio, riff, random noise -- while working
just on reading rhythms. That way when it's time to sight-read you are better able to make the rhythms work with less attention and you only have to focus hard on the other six things on the page.
c) Please indulge me in a digression. IME big-band bass parts are an unique musical experience because almost all the time they are written by people who don't know jack about how to play bass generally, how play bass in a big-band in particular, or how notate a part which will deliver whatever half-baked concept they may have. Most of the time if you see slash-chords there are a bunch of details the arranger really wanted but left out. Most of the time if you see a written-out part it's 90% unhelpful and unmusical, and your job is to craft a part on the fly while also picking out and nailing the other ten percent.
What you're supposed to do is read the chart, listen to the whole band, listen to the bass trombone, baritone sax and piano left-hand, and bring all of your experience to figure out what is going on musically, where it is likely to go and how you can build a part that will help bring that to fruition. And you're supposed to be swinging like crazy, pushing just hard enough so that everybody else has an easier time swinging too.
At its best it is unique, beautiful and joyful (and if done successfully no-one will notice). At its worst you're leaning over the porcelain throne wondering if they will let you leave early and run with no bassist.
All the best with your quest.