Does generalization just happen?
Of course not! Everything takes practice. This is why you should work with a teacher to practice speaking in situations that are increasingly difficult. You may start with something easy, such as learning how to pronounce the sound that you cannot produce or words that have that sound, then you may graduate to pronouncing that sound in sentences and eventually you have conversations outside of working with your instructor, such as with colleagues or with friends. This increasingly complex practice is called a hierarchy. More success generalizing strategies means speaking with a Standard American English accent more automatically, allowing you to speak clearly, not just with your teacher, but with everyone else.