Control Particle System's Seed using drivers?

I am creating a series of randomized scenes for a D&D Battle Map project. I would like to do as little effort as possible, naturally, so I am trying to generate a few instances of the same scene using different seeds from my particle system to avoid tiling.

Of course, I’d rather use a DRY method than simply keying a new value every frame, especially if I make this file a pipeline piece. My theory was that I could use a simple driver to control the seed and be done with it, but unfortunately that isn’t working how I would like it to. When I use this method, or even keyframing the seed by hand, it reinstances the particles but at the same position. The scale of the hair particle changes if I have random scale enabled, but still the position is identical between frames.

Any ideas?

Edit: What it is currently doing.