If I had a graph $G$ with some negative edge weights, clearly Dijkstra's algorithm does not definitely halt, since it might get caught in a negative cycle (shedding infinite weight). However, would finding the minimum weight (most negative weight) $w$ and adding the absolute value to every edge's weight preserve the shortest path, and get rid of the possibility of a negative cycle in one fell swoop? I can't seem to find any good literature on this, which makes me think that it can't be true.
Your idea does not work. Adding an absolute value to every edge wont preserve shortest paths. To see this take this graph:
u------v | | | | a------b
with all edge weights 1 except for $uv$ where the weight is 4. The shortest $uv$ path goes via $u \to a \to \to b \to v$. But if you add a +2 to every edge, the sortest $uv$ path is $u\to v$.
There is however a reweighting scheme that works. Check out Johnson's algorithm, which is built around this. Simply speaking: add a dummy vertex $x$ connected to all other vertices with a zero weight edge and add to every weight of an edge $(i,j)$ the value $d(x,i) - d(x,j)$.