@sky123,
I just responded to another thread here where I used the singular "they" without thinking about it. It made me laugh when I noticed I had used it.
I think I alternate between using "he" as a gender-neutral pronoun, and the singular they. I do this without thinking. I don't think I can tell you the pattern (maybe if I think more about it I can figure it out). I think I sometimes will write "he or she", but this is generally when I am specifically trying to avoid offending someone (i.e. at work).
In truth I don't worry about it or think about it very often.
That's not very helpful advice.
Hopefully you have found Americans to be pretty easy-going and not too uptight about these things. I can't speak for all Americans, but generally I don't make a big deal about slip ups, particularly with people I know are English language learners.
I speak a second language too, so I understand how difficult it is to get the phrasing correct.