The 142nd Open starts on Thursday at Muirfield, one of three male-only member clubs on the tournament's rota, and the R&A has again come under pressure to take action.
"Obviously the whole issue of gender and single-sex clubs has been pretty much beaten to death recently," Dawson said.
"And we do, I assure you, understand that this is divisive. It's a subject we're finding increasingly difficult, to be honest."
Dawson said the R&A had been at pains recently to try to explain some of the facts around the issue.
"Single-sex clubs are in a very small minority in the UK," he added. "Half of them are women only, half of them are men only. They're perfectly legal. In our view they don't do anyone any harm.
"The media are, with seemingly boundless energy and enthusiasm, giving out the message that this is an issue and that such clubs should be condemned to extinction and we shouldn't be using one to stage the Open Championship.
"We understand that view," Dawson said.
Nearly half the questions Dawson was asked in a 40-minute news conference were related to the issue of single-sex clubs, including one about the decision by Scotland's First Minister Alex Salmond not to attend the tournament.
"I think at the R&A we've been through over 250 years of existence without getting into political comment, and I don't really intend to break that rule here," Dawson said.
Dawson was also asked what, from a moral point of view, was the difference between men-only and whites-only clubs.
"Oh, goodness me, I think that's a ridiculous question, if I may say so," he said.
"There's a massive difference between racial discrimination, anti-Semitism where sectors of society are downtrodden and treated very very badly, indeed. And to compare that with a men's golf club I think is frankly absurd. There's no comparison whatsoever."
Dawson would not be drawn on whether there had been any discussion with Royal Troon and Royal St George's, the other two male-only member Open venues, and Muirfield about changing their policies.
"I'm absolutely not able to say anything more," he said.
"We do believe that membership policy is a matter for the clubs. We happen to believe that very strongly."
Dawson said the R&A would be discussing the situation again after the tournament at Muirfield.
"We will have a good look at what people are saying and try to take a view about all of this and find the most sensible way forward," Dawson said.
"We understand it's a polarising issue. We've been through other polarising issues - recently with anchoring, for example - and you eventually come out with a conclusion.
"I'm absolutely not going to pre-empt what's going to come out of this. I wouldn't even want to call it a review, but we're very conscious of the disparity of view on this subject."
Former world number one Rory McIlroy said he had never really thought about the issue.
"I just don't think it's something that is a real issue anymore," he said.
"Obviously it's an issue in some golf clubs. But in terms of life in general, I think men and women are treated equally for the most part these days. And that's the way it should be."
McIlroy said the players had not been advised about whether to discuss the situation.
"I just think it's something that a lot of guys don't want to get themselves into because it's quite a controversial issue," the Northern Irishman said.
"It's something that shouldn't happen these days. It's something that we shouldn't even be talking about."
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