Paths in a weighted line arrangement
The answer is no, me think.

To see why consider two lines that are a perturb copy of each other that intersect at a point p. By giving one line weight 1, and one line weight -1, we created a tiny wedge where the path can not go through. Think about this wedge now as a ray. Now, using a collection of such rays, we can create the following configuration:


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That is - interleaving horizontal rays. Now, the path must wiggle between these rays in the natural way. It is now trivial to add an almost vertical ray that its tail (i.e., non ray part) intersect all the horizontal rays. The path must intersect the two lines defining this ray as many times as we want. 

Here is a figure:



(source: uiuc.edu)  


(Ipe source.)

The solution is slightly more complicated because these rays open up. By adding the two rays on the sides, one guarentees that only the region of interest is really relevant.