Keep working towards your degree. This will help you in the future, but for now you will have a harder situation; perhaps. Most parts of the world prefer teachers to have a degree, of course. Some places don't insist, but you will usually have a choice of the lower quality jobs.
China hires teachers without degrees - sometimes. In Europe, most schools insist on degrees, but they are not required by law. Really, every country has different regulations and a different situation. You need to check the situation in every country to find this out.
Having a paid airfare is not always so good. Sometimes it's better to just go and search - assuming you have the funds to do so, of course.