The South not only did not trust Lincoln, they hated him. Real hatred. He was not new to the political scene, having been a vocal free soiler in the House of Representatives. While he was pragmatic about abolition, and had other priorities in the short term, the election of 1860 was a litmus test for the country on slavery and states rights.
Remember, it was impossible for people to vote for Lincoln in ten southern states, because his name didn't even appear on the ballot. When he was elected anyway, southerners pointed to the results as final proof that the North could dominate the South, and would eventually get rid of slavery altogether, if not now, then soon. From their perspective, how could they remain part of a union where one segment of the country could dictate a way of life to the other?
Their fears had been building for decades, and Lincoln's ascendancy to the Presidency was just the last straw for them.
Have Americans ever trusted what politicians said? Especially if they already do not like that politician? If you think about the last decade or so, you will know that we do not tend to believe what politicians say, particularly if we don't like them. Liberals didn't believe Pres. Bush when he said he wouldn't try to take away people's civil liberties. Conservatives don't believe Pres. Obama is really Christian or that he is not trying to make the US socialist.
In the same way, the South did not believe Lincoln. They thought that he was saying one thing in order to get elected and that he would then turn around and do the other thing once he was in office. So it all comes down to trust, and they did not trust what he said.