Money for education is exempt from the gift tax. There is no limit. You do not need to file a form 709. Reference: https://www.savingforcollege.com/article/tuition-gift-tax-exclusion
Using money from your IRA for education, is not exempt from income tax. It is, however an exception to the 10% early withdrawal penalty. Reference: https://www.investopedia.com/ask/answers/082515/can-my-ira-be-used-college-tuition.asp
Q. So do I just not include them on my taxes?
A. No. Those 1099-Rs are taxable income. The TurboTax interview will guide you on claiming the penalty exception on form 5329.
Q. Do we owe taxes on this money?
A. Yes
Q. It seems I can file a form 709 to avoid the taxes (but it lowers my lifetime exclusion) - am I correct here?
A. No.
It appears your advisor mixed up the rules or you misunderstood what you were told.
