To be honest, this isn't really my area as a mechanic, but I'll help as much as I can. September on the Black Sea is actually a great time - the water's still warm and the tourist crowds are starting to thin out. If you want to avoid the hustle and bustle of Sochi, I'd suggest looking at Gelendzhik or Anapa, just not the city centers. I've been there a few times myself, and there are some cool little towns like Arkhipo-Osipovka between Gelendzhik and Tuapse, it's way more peaceful and cheaper there. The water's nice, the beaches are decent, perfect for a kid.
Another option is Tuapse and the surrounding area - it's generally always less crowded than the top resorts. Or check out towns like Loo, Lazarevskoye - also in the Sochi area, but really they're separate places, not as touristy. In September prices drop like 30-40 percent from summer rates, easier to find accommodation. I'd recommend booking your place in advance, but don't go overboard - early September you'll still find normal options. The main thing is to go on weekdays instead of weekends, it's completely different then.