Hi Desertdryad, welcome to HAZ. I think your best backpacking opportunities closest to Flag will be West Clear Creek. Check out the Tramway/Maxwell Loop: http://hikearizona.com/decoder.php?ZTN=683
It's a moderate trip with extended dayhiking options full of breathtaking scenery. You'll also be camping next to a large perennial creek, so water will never be an issue, as long as you filter at least a few liters for the hike out.
The turnoff to the trailhead is less than an hour outside of Flag, although the dirt road that takes you to the trailhead will take a bit of time, especially depending on your vehicle. A few years ago I took my 2wd, medium clearance Tacoma there, and although it made it, there were some very sketchy parts. If your vehicle can't make it, just tack a couple miles onto the hike for walking the roads (which are also very scenic). This isn't necessarily a bad thing, though, since the loop itself isn't very long.