No they didn't.
Yes, they did. I already cited where Virginia opted to meet with Lincoln prior to signing the articles of secession. Lincoln refused. Then Lincoln doubled down, and Virginia joined the rest of the Confederacy.
The South offered to negotiate the terms under which they would leave the Union, the division of property if you will, with the implicit understanding that if the terms weren’t acceptable they would “appeal to arms,” as they frequently put it. IOW, it was a negotiation at gunpoint.
The South did NOT offer to negotiate WHETHER they would leave the Union. The Deep South considered the mere election of Lincoln to be so offensive they could not remain in the Union no matter what he said or did or agreed to. They were very clear on this point.
One of the issues that never seems to be discussed is that of division of the territories. The South had very specifically and intentionally fractured the Democratic Party over the issue of access to the territories, and now we are supposed to believe they would just accept the confining of slavery to the existing southern states, the whole issue over which they had fought for the last decade?
It seems much more likely to me that had the northern states agreed to allow the erring sisters to depart in peace, they would have started demanding a division of the territories, and quite possibly access to the Pacific via California.
IOW, I think a powerful group in the CSA had demands in mind that would have made war unavoidable short of utter and abject surrender by the Union. Of course, the same group had delusions (and that is exactly what they were) of the South conquering a slave empire at least to Panama and possibly Tierra del Fuego.