An ESL endorsement on your teaching cert would be helpful for teaching abroad, but it won't hold the same value as an MA of course.
MA TESOL is the ideal degree for teaching English abroad, but it certainly isn't the only degree. Any master's in applied linguistics, education, or even English literature would be highly regarded as well. An M.Ed. in ESL curriculum / instruction sounds like a good choice for you.
CELTA or an equivalent (Trinity CertESOL) means quite a bit abroad, but means nothing in the US. Nonetheless, I highly recommend doing a CELTA just to better yourself as a language instructor.
As far as experience goes, international schools would provide the most relevant work experience if you want to go back to teaching in the US later on. Teaching in a hagwon or public school scheme is better than nothing, but still means very little.
Good luck!