Do not cut them back now!
First-- what kind of azaleas do you have? Generally speaking, they are going to be the small, low-ish to the ground, with small blooms about 1-2' height kind OR the much larger (getting a big as 6-8' tall) with very large
flowers kind.
In any case, if you had no
flowers this year, it's not because you are having a drought THIS year. Azaleas bloom on last year's growth-- so it's last year's weather & environment that mattered. This is the reason one prunes azaleas right after they bloom, so that the plant can grow new growth for next year's blooms. As far south as Georgia, I'd guess you shouldn't prune after July 1 at the latest (this might even be too late). Point is-- if you prune NOW, you are sure to have no
flowers next year, either.
I take it that by "suffering" you mean that leaves are turning yellow and falling off. It's been a while since I lived down south & had azaleas, but I wouldn't blame it all on the drought. A stressed plant is more prone to pest invasions and disease. Be sure that they are mulched well (pine straw), and that you're soaking them with water every week or so. Beyond that, I suggest you check out Georgia's Ag Extension office. Usually when there's a state-wide problem, they will post something on the web site for the home gardener. If there's nothing-- why not contact them? They would be super familiar with what's gong on locally. Alternatively, do you have a garden club in your community? Or a local Master Gardener? They too should be able to offer you more concrete advise if no one here is able to.
Good luck. I miss azaleas!