If you haven't been backpacking in S Utah yet, consider yourself lucky that the best is still ahead of you. April-May and again mid-Sept through Oct are the optimal months.
It doesn't meet your goal of making good use of your free ticket if you're going from California.
I've hiked a fair bit in many locations in S Utah (14 trips of 7-10 days each) - Paria Canyon, Escalante, Dirty Devil, Dark Canyon, Grand Gulch. And many day trips in Zion, Arches, Canyonlands, etc. We've barely scratched the surface - we've never even been to the San Rafael Swell :)
There are NO BAD CHOICES in S Utah. Don't get frustrated if you can't get a permit to a popular place -- there are plenty of options. Get Steve Allen's Canyoneering 1, and Canyoneering 3 books and find something that meets your skill level and duration. If you've got climbing skills, and AFTER you've spent many weeks "below the rim", get Canyoneering 2.
Good luck, Amy