A sibling comment linking to Johnson's algorithm shows a way to do it.

To me, adding a constant to all edges seems like an obvious idea to try-- but one issue is that it will distort the ranking for paths that have different numbers of edges. Ideally a normalisation would preserve the ranking of paths so that a path with minimal unnormalised weight would also be a path of minimal normalised weight. But ranking won't be preserved by adding a constant C to all edge weights, now paths with more edges will be unfairly penalised.