The South's major objection to Abraham Lincoln was that he was a Republican, a party formed on the express platform of preventing the extension of slavery into the territories. Lincoln's election meant to the South that the Republicans were gaining ground, and slavery would eventually be eliminated in that area.
The existence of slavery in the territories was a major concern because it was commonly accepted that these areas would one day become states. If enough anti-slavery states entered the Union, a constitutional amendment ending slavery could be passed and the "peculiar institution" would be doomed. It was for this reason that the South argued vehemently for the admission of Missouri as a slave state earlier.
As for Lincoln's pledge, he was sincere. Lincoln believed--correctly--that slavery was constitutionally protected. He pledged not to interfere where it existed because he believed he did not have the right to do so. In fact, Lincoln had supported publicly and in writing the Corwin Amendment which would have protected slavery where it existed. The Amendment had passed Congress, and Lincoln wrote to every state governor, including those in the South, urging its adoption. He believed this would preserve the Union. The South's objection was not that Lincoln would attempt to end slavery then and there; but that his election marked the movement of the country down a slippery slope which could only end with the eventual abolition of slavery by constitutional amendment.
The Southern states seceded even though Lincoln promised not to interfere with slavery because they did not trust him to keep his word. In addition to distrusting Lincoln, they also hated him.
Lincoln was elected without the support of any of the Southern states. His name was not even on the ballot in those states and yet he was elected anyway. This led the South to be certain that he would govern in a way that would hurt them. They felt that Lincoln would come into office and go back on his promises. They felt that he would have no reason to care about Southern "rights" because they had not supported him.
Overall, then, Southerners had not voted for Lincoln and did not trust him. They went beyond distrust to outright hatred as they thought that he was a "Black Republican" who would destroy their entire way of life. For these reasons, they did not trust him and seceded from the Union, regardless of what he said he would do about slavery.