I think any native-English-speaking American (and any person from the UK or Australia or New Zealand or Canada or...) knows that there are situations when a higher degree of formality is required and those when a high degree of formality would be inappropriate.
I write a different e-mail to the CEO than I do to my boss, which is still different from the one I sent to my coworker. I talk different to my child's principal than I do to my neighbor.
What do you think your teacher meant when he or she said Americans tend to be more formal?
As for your pronunciation question, are you asking whether pronouncing the G is more formal or less formal? Omitting the G is less formal.
Student or Learner