Don't play what's on the page. Drum charts tend to be the poorest written parts of arrangements.
I have been studying with a very experienced teacher for over a year and a half and have made it into my high school's advanced jazz band.
For the most part, lock with the bass and keep time. If the bass is playing in 4, then play 4 light bass drum strokes per measure. If the bass is playing in 2, then play 2 strokes per measure. This will provide a solid foundation for keeping the rhythm section together.
Ignore whatever crap is written as your "drum beat" on the music. Swing charts usually have basic beats written, so they're not much of an issue. However, Latin charts always have some convoluted rhythm written out for you to play. Do not play this. Arrangers tend not to be drummers, and usually don't understand how each part of the drum set works in these types of charts. I would recommend learning the cascara rhythm to start off. The Drummer's Bible has tons of Latin rhythms, and I would highly recommend it for expanding your arsenal.
So now you're keeping time smoothy. The next part is band hits. Look for "ENS" or "Ensemble Figure" on your chart. This means the entire band plays what is written, so you should play along too. As a general rule, a quarter note or longer counts as a "long" note, and is played with a cymbal/bass drum hit. An eighth note or shorter is a "short" note, and is played on the snare.
You will also see background figures in your music. These are usually played by only one section while the rest of the band continues to play. You don't have to play these, but if you do decide to play them, keep time going and just play them on your snare.
One challenge is keeping track of your place in 100+ measures of "time." Since you're comfortable with combo work, I'm sure you know how to "feel" the progression of 8 bars by listening to the bass. Most big band charts are in some 8-measure form, so you really only have to keep track of the sets of 8 measures that pass by. I usually photocopy my charts and highlight the end of every 8-bar set.
I know this is pretty brief info, but it should help you get along with sight reading.