It is certainly possible to get a job teaching EFL without a degree if you have a TEFL or TESOL Certificate. The Trinity Cert is also recognised by many good employers. However, your options will be more limited than those of a person who has a degree. Some countries (especially in Asia) do not employ teachers without a degree. You’ll be able to find out about this by looking at job ads. It will also be more difficult for you to find a well-paid job. With a few years’ experience, however, this situation will change.
As for finding a school that will give support for new teachers, you will have to ask about this at interview. Make sure you ask specific questions to find out exactly what sort of support they give and possibly how much support. I’m sorry I am not able to recommend good recruitment companies; if you post a question in one of the other forums on this website, other teachers might be able to advise you.
The difference between TEFL/EFL/TESOL: EFL is English as a Foreign Language, TEFL stands for Teaching EFL. TESOL refers to Teaching English to Speakers of Other Languages: this could be the same as TEFL but also covers Teaching English to people whose country has English as a second language (e.g. India). The techniques used and language points covered would not be the same as in TEFL.
A good idea would now be to re-read any material you have from your course and possibly get in touch with others who were on your course to find out about their experiences.