Use neither. On the one hand, this is the literal difference between the two:
I am an adult's English teacher
suggests that you are a teacher to one adult;
I am an adults' English teacher
suggests that you teach multiple adults.
On the other hand, these sentences are both awkward. Possessives tend to work less well when long phrases are involved. In this case, it can become less clear what the possessive is determining: is the subject you teach "adult('s/s') English" or "English"? There are a few strategies for avoiding the extra awkwardness of the possessive:
Use a noun adjunct. "I am an adult English teacher." It still has ambiguity, namely whether you are an adult who teaches English or whether you teach English to adults, but my top Google search results turn up job ads for the latter. That collocation avoids the possessive entirely.
Verb the noun. Make teacher into teach and explain what you do after that. "I teach English to adult learners" or "I teach English to adults" would explain what you do. A couple of my first search results link to TEFL teachers who describe themselves in exactly this way.
Distinguish your audience in a prepositional phrase. "I am an English teacher for adult learners" or "I am an English teacher for adults." If it is important you say teacher, this breaks up the information in a way that makes adults more proximal to teacher than English, and uses for to disambiguate the purpose.