Creating raster with distance to feature using QGIS?

If I understand your problem, you can create a raster distance layer and clip that output to your region polygon.

• Create a raster distance layer using the Raster > Analysis > Proximity (Raster Distance) tool in QGIS (some more info here) -- Using your shoreline layer as the input
  -- This will create a raster where each cell has a value of its distance to the shoreline - you can set a max. distance if you want to cut off the analysis

• Then, use the Raster > Analysis > Clipper tool -- This will allow you to clip and Input file (Raster) by a mask layer (choose the region polygon)

• The output would be your region polygon as a raster with values of the distance to the shoreline.

Is that what you're after?