No. The Posi-Loc has no effect on whether or not the front diff spins both wheels. It just replaces the actuator so the front diff is engaged on demand with no waiting period. The carrier in the front differential is what determines how power is divided between the front wheels. Every factory front carrier on these trucks is open, meaning only one wheel will spin in a situation where you're losing traction. If you drive on the street at all, you don't want the front end locked. If you put a locker in the front diff the truck will not steer well at all on pavement with 4WD engaged.
That said, I'm assuming your truck is a half ton due to the six lug wheels in the pics you posted. That means you'll have the 8.25" front differential. No one that I'm aware of even makes a locker or limited slip carrier for that differential due to the weak cases. If you had the bigger 9.25" front diff there are some decent options.