I highly second what MisterStretch and Straydog have said. There is a huge distinction between most international schools and other types of schools here in Thailand, and a corresponding difference in the ways in which they recruit. The recommendations you'll see on Ajarn tend to apply to the latter since they're focused on English language teaching and teachers who are not actually trained or licensed (excluding a CELTA/TEFL/etc.). They do not apply to international schools.
If you're a qualified teacher with experience, you should begin your search far in advance. The best international schools like NIST, ISB and BPS have already wrapped up most of their recruitment for 2018-2019. Others will continue through March to April at the latest. In other words, you need to apply months in advance, and no reputable international school recruits through Ajarn or other ESL-oriented sites.
You should be applying directly through the listings on the schools' sites, and potentially applying to listings on TES, Search Associates and ISS. From what I can see, TRI doesn't deal with any of the better-known schools here. If you don't see much listed on these sites now for Thailand, I'd again echo what the others said: delay coming here and start looking for openings as early as this November for the 2019-2020 school year.
The difference in the work environment and pay will be massive. An average English teaching position here in Thailand ranges between 30,000 to 45,000 baht per month, with no other benefits - no housing, no flights, and minimal insurance and support. Class sizes can be quite large and the work environment can frequently be frustrating due to poor communication between the Thai and foreign staff.
With your background, you could probably arrive without having a job lined up (if your timing is right) and get a better position at a decent bilingual school. But even then, the pay will probably be in the neighborhood of 40,000 to 60,000 baht, as very few go above that. There's a slim chance that they may provide a small housing allowance of a few thousand baht, and maybe an annual flight if you're very lucky.
However, when it comes to international schools (excluding the smaller ones), there is a lot more potential. Most mid-sized schools having salary scales that start around 50,000 to 60,000 (at the low end), and more often 70,000 to 80,000. They usually have a housing allowance, and almost always provide an annual flight, decent insurance and PD funding. The best schools (those mentioned above) all start around 150,000 baht per month at the low end and go much higher, particularly when you factor in the annual bonus, housing allowance and other perks. Almost all of the schools in the middle and high range tend to recruit from overseas - which is again why you should ignore the advice on Ajarn if this is what you're aiming for.
Take your time, build up a little more experience, work on your CV and start applying at the end of this year. It's worth it.