Most of the best backpacking is in eastern Tn. This time of year, its hot, so you'll want to be high or on a cool creek. These would be my top picks:
The Citico/Slickrock Wilderness fits pretty well; its on the Tn/NC border. Its a lot like the Smokies but fewer rules. There are some nice high balds and great creeks as well. Lots of old growth forest there. I think dogs are fine, but you might want to check.
Smokies would work well, but I don't think dogs are good there. And its a bit more crowded.
Roan Highlands area is a great. Its high and has long stretches of balds. Its more northeast Tn. The AT runs through it. The rhododendron may even still be in bloom up there. Great walk with lots of views.