Cool story. If the DNA pans out it would provide closure for a mystery that has people mystified for better than 70 years. On such an island water would be a challenge to obtain in any significant quantities. Without any rainfall, and lacking suitable equipment for extracting fresh from salt water, it is doubtful they could last more than a few days.

If sufficient numbers of good sized fish could be obtained there is a way of sucking the water from them, generally you cut a notch through the spine behind the head and suck, but it is not likely that anyone in 1937 knew about such methods.

This was a time before reverse osmosis and without suitable vessels for boiling salt water and condensing fresh the output would be minimal.

A solar still would be an option, the Plexiglas from the plane would be suitable and vegetation or dark salt water soaked cloth would provide vapor, but I doubt many people knew about such methods in 1937.

Not every situation is survivable.