The neutral grip (palms facing one another) is the best grip to use if overhead pressing exercises are bothering your shoulders.
With the elbows pointed slightly forward instead of straight out to the sides, press the dumbbells using a neutral grip. Bring the dumbbells close together at the top without touching them.
I'm going to describe a little 6-week program that I call Tsunami Training (I'll get to why I call it that a little later). It's based, for the most part, on training methodologies that I picked up from Ian King and Charles Poliquin, with some razzle-dazzle thrown in by myself.