It is -dstnodata you need, not -dstalpha:
gdalwarp -cutline -dstnodata -9999 INPUT.shp INPUT.tif OUTPUT.tif
gdalwarp sets the area outside the -cutline polygon to -dstnodata, which can be the provided value (or one for each band), the nodata values from the input data set, or some default value. I don't know how GDAL determines the default nodata values, but for Byte data, it is zero.
An alternative solution that was proposed here "is to use gdal_translate with -projwin to extract to shape envelope and then gdal_rasterize with -i to burn nodata where shape is missing." This solution also circumvents raster shifting or resampling.
But all of these are using GDAL command line tools.