Some Southern Whigs remained Unionist during the war, while most supported the South. It's important to note that many Black Belt counties traditionally supported the Whig Party. A county-by-county study of 1860 election returns will show strong support for Bell-Everett in traditional Whig strongholds in the South.
Douglas was unacceptable to most Southerners due to his position on slavery in the territories. That of course is the basis of the "states rights" argument made in 1860, that citizens of all states should enjoy property rights in U.S. territories. Douglas held that people of the territory could vote slavery up or down prior to statehood; most Southern Democrats held that the states jointly owned the territories and rights to slave property could not be abbrogated prior to admission as a state.
A reading of Alabama newspapers will show that Douglas gets beaten up far more often than any other candidate, including Republicans. However, Douglas fared better in the election in Alabama than he did in neighboring states like Georgia, Florida or Mississippi. Tennessee went to Bell, her native son.