Add one more vote for the polyurethane glue. If the glue begins to turn a lighter color while you're spreading it out, then you don't need to apply any water to the core - it's already beginning to cure.

Edge gluing the sheeting is also unnecessary. As long as the edges are trued and taped together, they will "pinch" together when you apply the sheeting to the core. Then they get glued together with the poly glue that is attaching them to the core. Saves you alot of time!!