I'm not sure about the necessity for a degree. I wouldn't worry too much about it. The Trinity is similar to a Celta right? Why don't you search for ECC on the web and give them a ring. They have loads of language schools. A CELTA or equivalent would make you more useful than someone with a degree and no understanding of teaching methodologies. Looks are very important in Thailand and schools are often a bit shonky and low paid but a nice lifestyle.