I don't teach TESOL, but I know many people who do. They're all NT, but more introverted/open-minded NTs than most. I can't offer any advice on teaching abroad, but TESOL in this country can be a good field to work in. I've taught maths to TESOL learners, and it was a decent experience. If you want to go into teaching, I can think of a lot worse. If anything, the course will be interesting if you like languages.
As long as you communicate clearly and the classes have a very clear structure, the students aren't as judgmental as they are in other subjects. You might teach teenagers at first, but even this isn't as hard as teens in other parts of FE. The other teachers are usually well-educated and tolerant because they work with people from different cultures.
If you're in the UK, full-time and permanent TESOL jobs can be hard to get just after you qualify. People tend to get those jobs and sit on them. If you work for a teaching agency for a while to get more experience, it's easier.