Student or Learner
Do we say by way or by the way when we start a conversation?
"By the way I got a promotion!"
You don't use it START a conversation. You use it to shift the conversation to a new topic.
It's "By the way."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.