I had a promotion within the company this past year. Does that count as "Got a new job" when asked "Life happens. We're here to help. Any big changes last year?"

No, you would not enter having a new job.  The question refers to having a new employer.

No.   Those questions are preliminary to the ones you will answer on the income screens.  If you are still working for the same employer then you are not going to have a W-2 from a different employer---which was the point of that question.   Say no to "new job."