skassel - I disagree with your advice. The IRS won't know there is any amount owed to them until the prior year amendments are done. It depends on exactly what TP wants to have happen.
If 2017 is filed first, and refund from 2017 is applied to 2018, then taxpayer will owe the amount due when the prior year amendments are filed.
If the amendments are filed first, then the 2017 overpayment will (hopefully) get applied to the prior years first. Then TP would (presumably) need to make ES pmts for 2018, if willing and able. In this case, I would NOT mark the 2017 return as "apply refund to next year" NOR as a direct deposit, but would instead leave it as a paper check. If papercheck arrives, TP can void it, send it back, and ask IRS to apply it to prior year amounts due. (For some reason, I have noticed that IRS is more likely to apply overpayments to prior years when a papercheck is involved, then direct deposit. Don't know why.)
From a cash flow standpoint, under the first scenario, TP has to pay on the old years while not getting as immediate a benefit from the 2017 overpayment. Under the second scenario, TP will (probably) get a benefit from the refund used to pay off the prior years, but then has cash flow options about when to make ES pmts on 2018.