Mr. Dell’Aquila said he had not spoken to Mr. Cox about the matter and had not seen him since a fund-raiser last year.

The N.R.A. is moving on from Mr. Cox and is expected to announce Tuesday that Jason Ouimet, a deputy at its lobbying arm, will assume Mr. Cox’s former post, according to a person with knowledge of the appointment.

The N.R.A. has been burdened by high structural costs and escalating legal bills as it copes with the New York investigation and a bitter legal fight with its former advertising firm, Ackerman McQueen. The N.R.A.’s member dues fell in 2017 to their lowest level in a half-decade, as concerns about gun control ebbed after Mr. Trump’s election, but they rebounded last year, increasing by a third, to $170 million, while contributions grew by 24 percent to $165 million.

Even so, the N.R.A.’s net assets fell sharply last year and the organization was forced to freeze its pension fund. It also took more than $30 million out of its charitable foundation in 2017; it recently increased a line of credit, backed by the deed to its headquarters, to $28 million; and it borrowed against life insurance policies taken out on top executives.

An avid hunter, Mr. Dell’Aquila, 58, was interviewed at his house in Nashville, sitting in front of a large stuffed black bear. He played offensive lineman for a year at Princeton before injuries ended his football career, and he once was featured in The Baltimore Sun for eating two 48-ounce steaks in one sitting at a Shula’s Steak House. He’s been a member of the N.R.A. for about two decades, and considers himself a true believer. (“It’s our Second Amendment, it comes after our First Amendment, it’s what allows us to be a free country,” he said.)

Mr. Dell’Aquila was listed in an N.R.A. publication as having pledged more than $1 million, making him one of a small number of contributors at that level; he belongs to the Charlton Heston Ambassadors, a group for those who make large pledges of support, named for the actor and former N.R.A. president.

In a series of interviews and emails, Mr. Dell’Aquila cited numerous concerns. He was troubled that a former N.R.A. president, David Keene, had been caught up in an investigation over his ties to Maria Butina, the Russian who pleaded guilty to conspiring to act as a foreign agent. He was disturbed after The New York Times reported this year that Tyler Schropp, a senior N.R.A. executive, had an interest in an outside company that had received $18 million from the N.R.A. He was also dismayed by a recent New Yorker story tying the N.R.A.’s former longtime chief financial officer to allegations of embezzlement at a previous job.