Hey Bam,
Welcome to the Saloon!
You've been told right. The peak hiring season is June and July; most schools will be wanting you to arrive by late August/early September. There will still be some jobs in September, but most of the cream will long since be skimmed.
You've chosen nice cities. Heard mixed opinions of Haikou; the really nice part of Hainan is more down around Sanya.
All of those cities, especially the last 3, are extremely popular and have considerable competition for jobs. Earlier birds will definitely get better worms.
I'm in Suzhou...great place, but most of the private schools here are real slime pits. Be very careful; happy to consult before you come if you'd like.
Other options...Dalian, Nanjing, Wuxi, Xiamen, Shanghai are nice places.
Don't get your hopes up too high on air quality. Most places are at best "bad" by Western standards. Cities right on the coast (Dalian, Qingdao, Xiamen, etc.) might be slightly better.
Most schools don't have dress codes. Sure, they love it if you wear a tie, but most won't make a case of it if you don't. Corporate jobs are most likely to require a tie. Some schools do require it, so be sure to check. EF used to require ties in some places- not sure if they still do, but you don't want to work for them anyway.
Sounds like you got a good eye-opening in Korea. China is the same
with different flies on it, I'm afraid. 'Cept Korea pays better.
If ya got more questions, bring 'em on!