A great way to practice reading jazz big band figures is to write out pages and pages, and then read them at various tempos. The first thing I would work on, though, would be to make sure you can sing the figures like a horn player would, or in similar fashion to a jazz scat vocalist (Sarah Vaughn and Ella Fitzgerald come to mind). If you sing them like drum figures (short and staccato) they won’t swing!
Here are some examples. When I work on them, I sing the figures as swing rhythms (even though you might not know the specific pitches), and just fill as though in a big band. And listen to the big band greats such as Mel Lewis, Buddy Rich, Harold Jones, Jeff Hamilton, Peter Erskine, Ed Shaughnessy, John Riley, and Steve Fidyk for reference.