You can relocate azaleas any time of year when they are not in active growth (the period immediately following bloom) or the ground is not frozen. But if this isn't an emergency, I'd wait until the weather is cooler and you are a little closer to Fall rains...moving them now is going to require extra attention to watering, possibly even daily.

The plants shallow rootball will extend out as far as the branches reach, but if weight is a consideration you can take a smaller rootball safely. Aim for a rootball with a diameter equal to two-thirds the plants height. Drag on cardboard or a tarp to the new site if they are larger than wheel barrow size. Unless quite small, don't lift by the main stem, weight of soil and root ball could be damaging to roots.

Don't plant any deeper than it was originally growing, allow for settling if moving to a newly worked bed. Water well, keep moist, mulch.

Thanks for the advice. :-) It's not an emergency, so I'll let it wait till September. The bushes are still small, they came with the new house and the builder planted them in the spring, I think. They're in full sun and definately not happy there (they tend to wilt, even with watering, and two have died), so since I wanted to redo that bed anyway, I figured I'd put them in my future shade garden, where they'd look nicer and might do better.

If you are in Georgia, I'd wait until the second half of October or even November.