Finding the two node-disjoint paths, minimizing the sum of their lengths
This can be solved by a small modification to Suurballe's algorithm, with $O(|E| + |V| \log |V|)$ running time.  You might also be interested in Computing the k shortest edge-disjoint paths on a weighted graph.
Suurballe's algorithm works with directed graphs.  You can convert an undirected graph to a directed graph by replacing each undirected edge $(u,v)$ with two directed edges $u \to v$, $v \to u$.