Honestly, both of those sentences sound strange. The first, "with sweat" makes it sound like sweat is an accessory or a partner. "In a sweat" is a more figurative expression that means "worried". To describe someone's state while running, it would be more natural to simply say, "She was sweating while running." although even that sounds a bit redundant (of course she's sweating). "Worked up a sweat." is a good expression for this situation. "She worked up a sweat running."

As usual, I'm with Peachey on this one. I think that this has nothing to do with the usual sense of 'run' . The girl isn't running anywhere. I think the writer is using the set expression 'running with sweat'. It means that she was so hot, so stressed or so nervous that the sweat was pouring down her.

"Running with sweat" makes me think she was sweating so heavily, it was running like water. (No, it's probably not "running" as in moving your feet.) Using "in a sweat" instead doesn't sound natural or even similar to the first sentence.

I suppose that the sentence "She is running with sweat." could actually be "She is running with A SWEAT SUIT". Otherwise, "She was sweating while running."(like Johnny said) OR "Running made her sweat" OR "She started to sweat wile running" would be more appropriate.