If you intend teaching ESOL you have been given the right information, but that is not to say you won't be able to find a job with the qualifications and experience that you've got. Further education colleges normally require one to have a Level 4 ESOL subject specialist qualification or be working towards one, but you might still be able to find a job in a FE college with what you've got.
Have you thought about working for training providers. A lot of them are willing to take on new and/or inexperienced teachers and the teaching you'll be doing is virtually the same. A lot of training providers actually provide courses on behalf of FE colleges, so in reality the course is actually an FE one. But do choose wisely. Some training providers are brilliant and others quite frankly are absolute crap and just looking to make a quick buck at the expense of staff and students. Large training providers with locations nationwide tend to be the best for obvious reasons, but that is not to say small ones are not.
If you intend to teach EFL, then your qualification should be sufficient, but most EFL Lecturers/Teachers in FE have a DELTA or MA.