The advantage is 24Na's short (14.9 hour) half-life, not the lack of solubility in water. Half of the original radiation will be gone in 15 hours, and close to 90% in two days.
I understand half life. My point moisture currents can spread water soluble things very far very quickly.
Think of it this way, what happens if there's a catastrophic sodium leak? The winds carry it far and the Na gets into everything because it will be diluted into the wind's moisture. Won't you breath the radioactive Na from the air's moisture?
I imagine drinking only bottled water for a week, and perhaps increase the intake of salt for the same time. (Be careful if you have hypertension, it may be more dangerous the additional salt than the radiation.) Beer and salted peanuts looks like a wonderful anti-radiation plan.
If you get enough radioactive sodium salts to get covered in a dust layer, you are probably in trouble anyway. It may help that sodium is soluble an it can be washed easily.
If the small hidden sodium salts leak to water streams, my guess is that the concentration will be smaller than the natural sodium, and eating some additional non-radioactive sodium may help to remove it from inside the body even faster. Something like the potassium iodine pills.