For those who are good with Djikstra or graph problems

I have an assignment dealing with a graph problem that contains counties delivering surplus electricity to other counties that have a negative surplus. To add to that, my program has to run within a certain time and also the total transfers between the counties must be at a minimum. My question for now is: after I find a county with excess and a county with a deficit, what algorithm is there to find a the shortest path from start to end? I currently know of Djikstra's only, but that only finds the short path from a vertex to ALL other vertices. What I need is from one to another. Any ideas would be great! Thanks.

Oh yeah, if you do know of a way to solve the problem, it'd be great if you could explain as much as possible, since this program is also under a time constraint and due in a few days. I need all the time and help I can get.
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