Their conclusion was correct. You need to pay the expenses in the same year as the distribution. No problem with paying this year for next year's room and board.
You do not necessarily have to pay it directly out of 529 distribution money - you can use any money (or charge cards, or loans, etc.)
True, Publication 970 does not state that the expenses must be incurred in the same tax year. But it also does not state that chickens have feathers. As a cash basis taxpayer, you must account for things like qualified higher education expenses and 529 distributions based on when you receive and pay money (or money equivalents), at least until IRS specifically says otherwise.
[CMA: This is not legal or tax advice. You should rely on your own tax adviser.]