i have been to taiwan and their is westeners who teach without a degree, not legally of course. you can bribe officials or simply do visa runs. in taiwan a degree is not needed to teach but to obtain a work permit, regardless if you wanted to teach or do barwork. why anyone would want to live more than a few months there is beyond me! teaching english does not suit everyone. i would go somewhere and try it first. volunteer for a month see if you like it then if you do then great go ahead. an ex of mine still teaches in taiwan and the hours she works are 9-11 am 1-6 pm and she does evening classes twice a week and sat afternoons to make ends meet. she takes home around £1200 a month. the teachers in taiwan i met never really seemed that happy, i cant recall one that had real job satisfaction and most left their jobs and the country after a couple of months.
11

comenius
As you stated you want to work legally, you're going to have to finish your BA as neither Japan, Taiwan or Korea will give you a work visa without one.
Also, definitely do the CELTA if you haven't any teaching experience as it will (1) make you more attractive to employers and (2) give you a crash course in how to actually teach EFL.
ESL Jobs World: http://www.esljobsworld.com
Pro tip