list manipulation – Build Table of Edge Weight Sums

Is there a convenient way to build a table as generated by the code below where a 5th column (at the far right) could be added to the table that is the sum of the edge weights for the nodes in a given row. Then also be able to sort that table from smallest edge weight sum to largest edge weight sum?

Clear(edges,g)
edges={N1->N2,N1->N3,N1->N4,N2->N5,N3->N5,N3->N6,N4->N6,N5->N7,N6->N7};
g=Graph(edges,VertexLabels->"Name",EdgeWeight->{1,2,3,4,5,6,7,8,9},EdgeLabels->{"EdgeWeight"},EdgeLabelStyle->Directive(Red,20))
WeightedAdjacencyMatrix(g)//MatrixForm
TableForm(FindPath(g,N1,N7, Infinity,All))