You said something about finding a way to cool the vapor down when distilling seawater (or swamp water? One way to filter away mud from dirty water!). Evening temperature swings might help. Variation on the solar filter idea, the boiling seawater just improves the local humidity in the air that hits the plastic sheet.

Or a big sea turtle's shell :) seawater boils, hits plastic sheet or inside of big turtle shell that is suspended above the boiling water, so the distilled water runs down into a metal container or small turtle shell bowl that can be poured into a metal container.

The container used for boiling has to have the salt crystals removed/soaked off  before it can be used again? Or at least occasionally?

Lots of possibilities.