The notion that people lie to pollsters so they won’t look racist was called the Bradley Effect because of a poll about a politician named Bradley who didn’t get as many votes as had been predicted. People were sure Obama would get fewer votes than the polls predicted because of the Bradley Effect. He did not. Since then, people don’t bother to talk about the Bradley Effect.
I agree racists hate being called racists—both Derrick Bell and David Duke insisted they were “racialists” not racists—but they’re not hesitant to share what they believe is true.
Good polls like Pew have been very accurate. I was also skeptical about polls until I began researching them. In this case, I would trust the better-rated poll. They know their methodology matters, so they tend to provide it.