How naval guns handle fore-and-aft firing

How well do naval guns stabilize when firing fore or aft instead of broadside?

Usually well. More shock passes through the hull because more ship mass lies behind the recoil, but the mount is designed for that firing direction.

Naval gun mounts are designed to stabilize through nearly their full traverse. Firing fore or aft places more of the ship's mass behind the recoil than a broadside shot does. That geometry sends more shock through the hull, even when the mount remains stable. The additional shock is usually within what the ship can tolerate. Direction therefore changes the load transmitted through the vessel without making fore-and-aft fire inherently unworkable.