One dumbbell, full stability, pure shoulder.
Don't let your torso lean away. Keep everything stacked so the core resists the tilt.
Pressing one dumbbell overhead is a strict shoulder press with a hidden side quest, the offset load tries to bend you sideways and your trunk has to refuse. The delts drive the press, the triceps close out the lockout, and the free hand cannot help. It earns its place for building each shoulder independently and for teaching the stacked, braced overhead position that carries over to every press you own.
Your core is giving the weight a shortcut. Squeeze your glutes and brace your midsection before the press, and treat any lean as a sign to lower the dumbbell a notch.
It is a complement, not a replacement. The barbell lets you load heavier, while one dumbbell at a time fixes side-to-side gaps and adds a core demand the bar never gives you.
Standing is the default, since the anti-lean core work is half the value. Go seated only if you specifically want to isolate the shoulder and remove the balance element.
