What about the San Juan? Not much for rapids, but it is deserty.
If I read the regs right, you need to file an application, even though this year's lottery has passed. Once your application is on file, you can call in for cancellations and unfilled launch dates.
San Juan River
According to their launch calender their are unfilled launch dates the next three weekends.
Other trips: Grand Canyon from Diamond Creek down; Cataract, but you would need a motor or a jet boat shuttle to get it done in in 3-4 days.
Have you got a frame and oars for your boat yet?