I suggest you apply for some jobs that you see advertised and see whether you get any positive responses. If you get positive responses, you’ll know that your current qualifications are sufficient. If the responses are negative, contact the school and ask for feedback on why you weren’t successful. You can also contact schools that you would like to work for and ask for their opinion on your qualifications and suitability.
I’m not really familiar with American high school teaching diplomas. In the UK, people wanting to teach in state schools do a Bachelor’s degree and then a PGCE (Post-graduate certificate in education). A PGCE normally takes one year and is full time, the year is divided between study and teaching practice. When recruiting, many serious EFL schools consider the PGCE to be equivalent to a Diploma in teaching EFL. The diploma is at a higher level than the certificate (CELTA or TESOL).
I think you’ll be able to get a job with the qualifications you have. You can look at EFL schools as well as American high schools abroad; there are American high schools in many cities with a large expatriate population. If you choose EFL schools, you’ll need to be prepared to describe exactly what your qualification is. The advantage of the CELTA or TESOL is that recruiters in TEFL know exactly what it is and have an idea of what the applicant is capable of.
I think you should give it a go and make decisions based on the feedback you get.
Best of luck