The hiring season is usually spring, yes. Like from February to May. It varies from Uni to Uni. I'm sure many are still hiring, but like I said act fast. Each day that passes could also be an application deadline that passed.
Regarding the TESOL cert, mine is from South Korea (I used to teach there). I don't think my Uni even looked at it, and I didn't bother getting it notarized/authenticated (like I had to do with my Master's). I'd say it's nice to have, and definitely valuable if you lack a Master's or PhD. But with a PhD it'd probably just be the cherry on top.
I like my Uni. I got it because my wife is awesome and saw an ad on their website. I had two interviews, one with a department head, one with the entire hiring committee, and I had to do a demo lesson. The salary is not great (about 60,000 NTD a month), but I only have to teach 14 hours a week, there's no desk warming, the students are (mostly) mature and easy to get along with, I make my own curriculum, I get the summer months (and one month in winter) off, and let's be honest it's better for the ole CV to say you worked at a university than a cram school. Be warned, there is a lot more prep required for a Uni job. But it feels rewarding crafting your own curriculum and classes. It's not perfect, and there's some administrative headaches occasionally. And like I said the pay kinda sucks. But, I definitely wouldn't quit it. I feel lucky to have a Uni job here as they're in quite high demand.