I'm sure I'll be down-voted to hell for this, but to answer the question in your post title--- I wouldn't recommend Fukuoka as a "starter" city for Japan.
Language -- Far less English speakers than places like Osaka, Tokyo, Kyoto, Yokohama. If you'll be teaching English and don't already know Japanese it's going to be hard for you to learn on the fly, unless you're extremely self motivated or taking Japanese night classes. The English teachers I've met in Japan, even the ones who can speak Japanese well, tend to associate with other English teachers and seemingly don't settle into the culture here like people who work for Japanese companies and use Japanese at work.
I came with zero Japanese knowledge and it took be 6-12 months to feel comfortable using the language. I'm a notoriously poor students, but I did take full time classes (5 days a week) for 2 months in Japan and one month in my home country. I still study daily, though now it comes much easier and I can self-study. Do yourself a favor and learn hiragana and katakana before you arrive.
If you've lived outside the UK for a while, especially in another Asian city, you'll adjust quicker.
I know the feeling of wanting to leave a job and jump into the adventure of travel, but I highly suggest you get a job before arriving. It will help you get your foot in the door to other opportunities. You can even get a job, arrive, and then start looking for a better job while working.
You seem like you have your ducks in a row, and if you finished CELTA then I am sure you can live in Fukuoka and get by. I'm sure you'll be happy, life is good, weather is good, food is great, cost of living is more affordable than Tokyo. I'm just not sure I would suggest it as a "starter" city to live in if you've never lived abroad before. Smaller cities like Kyoto, Kobe, Yokohama, or even Nagoya, that are close to major hubs like Tokyo and Osaka might be better starter cities and still give you that "real Japan" feeling that you seemingly will prefer.
All things said, I feel like Tokyo is the best "starter city" to Japan. /end rant