I'm trying to be kind...but to teach English as a second or other language (ESOL) I would hope you had a good idiomatic use of English. Your written post shows that you don't have this standard yet. If you are contacting employers and this deficiency comes across in your written application, then you will not be their first choice. The purpose of an additional course would be to help you get to the required standard. I appreciate that your first degree may have given you excellent insight into English culture but you really need accurate, daily practised use of the language before you can claim to be proficient.
FYI and for anyone who feels I am being harsh, I have observed many instances of poor quality ESOL teaching done by unqualified workers, and this has the effect of further confusing the learner and preventing them achieving their aim of becomng proficient in speaking, writng and understanding English.