You are trying for a permit in the Prime hiking season. It will be tough.
You're providing no details about capabilities, intended routes, or duration. That makes it tough to help.
If you are strong, capable of route-finding, and not intimidated by an easy 20' (dead if you fall) climb, and there is no snow at the Tanner trailhead consider Tanner to Grandview along the Escalante Route. It's one of the best down there.
Figure out your itinerary. Figure your your dates, alternate dates, and how much you could "slide" earlier or later. Know how far you can walk in a day.
Call the Back Country Office around 2pm, when they're not to busy. Ask them if what you want to do is possible. Then ask them what IS possible. Offer a range of dates and durations. They will be able to run the dates and locations. If they can find something that works, fill out the forms - with as much flex as possible - and send in the fax, asap.
Also, if you haven't been hiking much, and you end up walking to the river on Day 1, consider making Day 2 a layover. It's not far, but it is enough Down to be debilitating to the unprepared.