Question

why did stephen douglas, among others, believe that “popular sovereignty” could resolve sectional divisions of the 1850s? why did the idea not work out?

Answer 1

Prominent power was Stephen Douglas' ticket to the White House. As a Northerner, Douglas needed to locate some political equation that would win him the support of southern Democrats without distancing every potential supporter in the North. 
To Douglas the answer for the subjection issue that had so stopped the Congress was well-known power - giving the voters of a domain a chance to choose for themselves on the off chance that they ought to be conceded as a slave state or a free state, in this manner assuaging the Congress from that vexing choice.