Sometimes if I want a small chamfer around the pocket I do an engrave cut first with a V tool, this leaves a nice edge but I only use hardwood and that always leaves a better edge than pine anyway.
A straight flute cutter would be better than the one you use at the moment or as suggested a down spiral generally works best.
These cutters give me good results at great prices; https://www.toolstation.com/shop/Pow...er+Bits/sd2579