Southern democrats of the 18th century were conservative to the bone. States rights conservatives. The Whig/Republican Party, especially the Republican Party of Lincoln, were big government liberals. Lincoln and Marx were great admirers of each other.
You may mean the 19th century. Democracy was still controversial in the 18th century.
If I could buy and sell people and force them to work for me without pay I wouldn't want the federal government messing with that either, especially if people like me controlled state and local government.
But Southern Democrats weren't opposed to using the federal government to protect and promote and extend slavery. They also weren't any great friends of freedom of speech or the press or contract or movement.
And those "big government liberal Whig/Republicans"? What did they want?
Federal aid for roads and canals. Protection from foreign competition for industry. Maybe the Homestead Act and land grant state colleges.
Not exactly a massive federal bureaucracy.
Nothing that Taft or Coolidge or Eisenhower or Reagan or Trump would have much trouble with.
Right and left, liberal and conservative can mean different things at different times and under different circumstances.
If those Southern Democrats had been able to make the country, the continent and the world safe for slavery, many Americans would have regarded that as a radical departure from our best traditions.
Also, you have to consider whether or not slavery was bad enough to be something you'd want to worry about -- whether or not it could be an exception to the idea that the federal government should keep its nose out of local and state affairs.
If I'd been around in the 1850s, I might well have supported the Democrats thinking that they were the more conservative party.
But I would have been wrong.