The invitation made an extraordinary offer: Donors willing to write a check for $500,000 to $1 million would be granted access to Donald J. Trump the day after he is sworn in as president, along with the opportunity to participate in a multiday hunting or fishing trip with his oldest sons, Donald Jr. and Eric.

But on Tuesday, the Trump Organization put out a statement distancing the family from the fund-raiser, saying the event had “not been approved or pursued by the Trump family,” even though legal documents show that Eric Trump served on the board of directors of the recently formed charity behind it, the Opening Day Foundation. The event is still scheduled, but references to the hunting trip have been removed, and Eric Trump said late Tuesday that neither he nor his brother would attend.

The abrupt turnabout was the latest example of the ethical thicket the president-elect and his family face as he prepares to take the oath of office. It highlights the need for Mr. Trump to clearly define what roles his adult children will play in his administration, according to former senior White House advisers who have served the last six presidents.

“I am seeing insensitivity to what is ahead,” said Michael H. Cardozo, who worked in the White House counsel’s office under President Jimmy Carter. The Carter administration struggled with the activities of the president’s brother, Billy, who had business dealings with Libya while Mr. Carter was in office.