I'm no expert, but it might be useful to consider how Mars no longer has flowing water on the surface, though many believe it once had lakes and oceans. The change didn't occur because of overheating, of course, but because of a change in atmospheric pressures. Not that HPL would have known that!
But Martian water is trapped at the poles in the form of polar ice. If the earth suddenly dried up for this reason, we could still get at the water. In fact, we wouldn't even need to move to get it. We could just mine it and de-salinate it and pipe it to the spots needed.
I'm not sure if we understood what the Martian poles were at the time of this writing, though.