If you are staining wouldn't it be better if you used veneered mdf , might be a bit dearer but at least you will be able to stain it and get the finish you are looking for.
I use mdf and I usually paint it which gives me a good uniform finish and covers up any unevenness.
For the edges well, as you say "the cut edges soak up paint like its going out of fashion", don't think there is much you can do
Paint usually fills it so maybe a few thick coats of clear may do the same.